Torsion free arithmetic

In the previous post, we noted that natural arithmetic $(\mathbb{N},+,*)$ is aperiodic in both addition and multiplication. As aperiodic operations they are of course directly related to partial ordering relations. The addition operation is directly related to the standard total ordering and the multiplication operation is directly related to the divisibility partial ordering with zero as the maximal element. Both of these are essentially distributive lattices as partial orders, which are also multiset inclusion lattices. The operations are multiset combination because natural addition and multiplication are both free. That is the situation with natural arithmetic. In the case of the integers $(\mathbb{Z},+,*)$ neither addition or multiplication is aperiodic, so we have to consider the related property of being torsion-free.

Torsion-free addition: the addition operation on the integers $(\mathbb{Z},+)$ is torsion-free. The reason for this is that all the negative integers are simply mirror images (or algebraic inverses) of their positive sides and so they still have the same infinite torsion-free behavior. So this means that the origin of periodic arithmetic does not lie in addition.

Partially torsion multiplication: the multiplication operation on the integers $(\mathbb{Z},*)$ is essentially the same as the on the natural numbers except that there are two elements ${-1,1}$ which are in a product with the ordinary integers. Furthermore, these two elements ${-1,1}$ form a cyclic group $C_2$ which means that $-1$ alone has periodic properties among the integers and only with respect to multiplication.

Natural arithmetic is nonperiodic

An aperiodic semigroup is a semigroup that contains no non-trivial subgroups. This means that for all elements $x$ there is a point at which further iteration of the element produces no effect or in other words $x^{n+1}=x^n$. Elements of this form a clearly aperiodic, and that is already established in the literature. A separate issue, however, is rather or not elements that can be iterated infinitely are considered periodic. It is clear that since these elements continue to infinity without any repetition they cannot be considered periodic either. As a result, a separate concept is nonperiodic commutative semigroups.

We can immediately see that $(\mathbb{N},+)$ and $(\mathbb{N},*)$ are nonperiodic. In $(\mathbb{N},+)$ the first element 0 is idempotent, and the remaining elements continue infinitely. In $(\mathbb{N},*)$ the first elements are 0 and 1 both of which are idempotent and then the rest continue infinitely. The only difference is that multiplication has more idempotents. As a result, natural arithmetic has no unorderly aperiodic behavior.

Non-periodic semigroups have the most order-theoretic behavior among the class of semigroups. Non-periodic semigroups can be defined by the composition of extensive monotone functions of a partial order. The partial order of addition is the natural partial order, and the partial order of multiplication is the divisibility partial order. The divisibility partial order is a suborder of the natural ordering except that zero is considered maximal. With respect to these orderings, we can see that addition strictly increases the natural ordering and multiplication strictly increases the divisibility ordering. Multiplication by zero simply transforms an element to the maximal element in the partial order.

It is also noticeable that the addition and multiplication semigroups are therefore related to semilattices. In particular, the maximum semilattice and the least common multiple semilattice. Considering these as upper bounds we see that $max(a,b) <= (a+b)$ and $lcm(a,b)| (a*b)$ as maximum is the least upper bound of its ordering and addition is a much greater bound and likewise lcm is the least upper bound of divisibility and multiplication is an upper bound that is non-minimal.

The nonperiodic and orderly behavior of the arithmetic of the natural numbers is the basis of the connection between arithmetic and logic. Both the natural arithmetic operations and the lattice operations are commutative, associative, and nonperiodic. This is why when we have two disjoint sets $A$ and $B$ and we take their union the cardinality is equal to the sum of the cardinality of the two of them. Addition is an abstraction of the operation of joining two sets in set theory and classical logic. It abstracts away the elements of a set and it tells us about their cardinalities. In the same way, multiplication is an abstraction of the joining of partitions in the co-partition lattice defined in partition logic. In this sense, arithmetic exists to benefit the understanding of logic.

Commutative aperiodic semigroups theory

The idea of commutative aperiodic semigroups theory emerged from considerations of generalizations of semilattices. The first aspect of this is that the idempotent property must be sacrificed in order to consider various generalizations that allow for repetition. Towards that end, we first consider commutative aperiodic semigroups which are distinct join preserving, which therefore are essentially equivalent to semilattices except for the condition that iteration and repetition is allowed. This is the general case until the commutative aperiodic semigroup of order 4, which is ordered by the weak order [2 1 1] appears.

Well considering this, another direction of thought came to me based upon the idea of bounds. The idea of a join and a meet are defined by the least upper bound or the greatest lower bound of two elements. But what happens if you relax the least or greatest condition? Then in that case you get a commutative aperiodic semigroup, so we can see how commutative aperiodic semigroups and perhaps even eventually other types of commutative semigroups can play a role in order theory.

In the place of semilattices we can instead produce a partially ordered set of commutative aperiodic semigroups on a partial order, determined by the leastness of the upper bound produced by the semigroup. This leads to the notion of a greatest upper bound, naturally this would seem trivial, but actually it isn't if we restrict ourselves to the condition that the result that the partial order is preserved by the algebraic preorder of the semigroup. This leads to our understanding of commutative aperiodic semigroups.

The trivial case: in the trivial case we know that there is only one possible semigroup so distinctions don't matter

The total order T2: there is only two cases the semilattice and the non-semilattice

The total order T3: in this case the commutative aperiodic semigroups with the total order on three elements can actually been ordered in a four-element diamond shape. The most semilattice like is the semilattice itself, then there are too cases that relax the semilattice property somewhat by making either the minimal element or the middle element a generator of its parent. The least semilattice like is the monogenic semigroup on three elements. The monogenic semigroup is essentially the greatest upper bound as compared to the least upper bound of the semilattice.

The tree order [2 1]: in this case the three types of commutative aperiodic semigroup can be determined by the number of idempotent elements they have. The more idempotent elements the more semilattice like the semigroup is. There are three cases the semilattice, the case where a single element is nilpotent with index two, and the zero semigroup itself which is the least semilattice like and which has two elements of index two which generate the zero element. This produces a semilatticeness partial order on the types of semigroups that have this algebraic partial order.

The general principle proceeds accordingly for the larger commutative aperiodic semigroups. The exceptional semigroup on four elements is the least semilattice like semigroup on the partial order [2 1 1]. The property of not preserve distinct joins makes a semigroup even less semilattice like, so it can be considered to be part of the hierarchy of different properties related to the ordering of semigroups based upon their semilatticeness.

Remainders from roots

I described how multisets can be divided to get a quotient and a remainder. This generalizes the process of division of a number, which is actually equivalent to dividing a equal multiset. The process of dividing a multiset can be generalized to other multisets though, which produces a different process. The first immediate thought is that this can be applied to prime factorizations. So for example if we prime factorize 24 we get {2,2,2,3} and if we divide it 2 we get {2} with remainder {2,3}. This division of the prime factorization multiset is essentially the same as taking roots.

So if we take the square root of 24 we get 2 with a remainder of 6, and it is expressed as $2\sqrt{6}$. I particularly like the number 360 because of its unique minimal prime factorization {2,2,2,3,3,5} which forms a progression multiset. If we divide this by two we get $6$ with a remainder of 10 so it is $6\sqrt{10}$. In the case of a cube root we get $2$ with a remainder of $45$ so $2 \sqrt[3]{45}$. In any case, the remainder is the object still in the root symbol and the quotient is the part which is not.

Ordinary division is essentially additive division so when computing division the remainder is added to the quotient as a fraction and the introduction of fractions is what distinguishes the result from the ordinary integers. Roots are multiplicative division so the remainder is multiplied by the quotient, rather then added to it. The remainder is then the algebraic part that distinguishes it from the other rational part.

I have seen the remainder be used to refer to the fractional part of a root, so the square root of 24 would then be 4 with a remainder of 0.898979... going on infinitely in a non-periodic manner. It is useful to consider 4 as the square root of the smallest square number less then the number, but it is wrong to consider 0.898979... to be the remainder. Instead this is the fractional part of the root. This is largely an issue of terminology, but it is interesting nonetheless.

Multiset division and remainder

The operations of division and remainder, so familar to us because of their use with the natural numbers, can be generalized to multisets. Numerous familiar concepts can be generalized and better understood through the use of multisets, which are relatively underused in the current situation. The first thing one realizes is that we can use this to divide and get the remainder of the prime factorization multiset of a number. So for example with 8 we can divide its factorization by 2 to get 2, and a remainder of 2 which is actually how we compute roots anyways to get $2\sqrt{2}$. So this means that nth-roots are actually based upon dividing a multiset, though it was never taught to me that way and I don't think its that common to describe it like that. Here is some clojure code that deals with the general case of dividing multisets.

(defn multiset-division
  [coll divisor]

  (Multiset.
   (into
    {}
    (for [elem (support coll)
          :let [n (multiplicity coll elem)]
          :when (<= divisor n)]
      [elem (quot n divisor)]))))

(defn multiset-remainder
  [coll divisor]

  (Multiset.
   (into
    {}
    (for [elem (support coll)
          :let [n (multiplicity coll elem)]
          :when (not (zero? (mod n divisor)))]
      [elem (mod n divisor)]))))

Distinctions among limit points

Firstly, I would like to briefly consider the different types of limit points and the nature of their differences. A topological space can be defined entirely in terms of its limit points, as the limit points are precisely the points generated by the closure of a set. In a way topological spaces exist to describe limit points. In typical applications, limits are defined in terms of the behavior of sequences as they approach infinity, so these limits are generated by an infinite process. The definition of a topological space, however, allows for limits that are generated by individual elements.

• Special limit points: limit points that are generated by a finite set
• Analytic limit points: limit points that are generated by an infinite set

The special limit points are generated entirely be the specialization preorder of the topology. This is really then about the relationship between order, topology, and the infinite. Topology only gains its own identity as distinguished from order theory when dealing with the limits of infinite sets. Alexandrov topologies are entirely generated by their special limit points and they are determined by their specialization preorder. Analytic limits arise in the purest sense in order topology when dealing with suborders of non-hereditary-discrete total orders (total orders other then $\mathbb{Z}$). These topologies are Frechet because they are generated entirely by analytic limit points, and metric topologies share this property. Hopefully, this clarifies the nature of topology.

Max order one multiset systems

In a previous post I described how the ordering of distinct max order one multiset systems is a disjoint union of total orders. This describes the order type of the support of a max order one multiset system but it does not completely describe the multiset system type of the max order one multiset system.

Therefore, we need to further consider max order one multiset systems, and the multisets of prime powers, which often emerge from commutative groups. Consider as an example {2,2,4,4,3,3,27} then in this case we can partition this by the support of each of these multisets and therefore we will get {2,2,4,4} and {3,3,27}. Then each of these has their own signature defined by the multiset of exponents of each of these multisets of prime factorizations in this case, though the multiplicative concept of a positive integer and a finitary multiset are equivalent. The multiset system type of these is then the multiset of exponent signatures of these components.

So for the example of {2,2,4,4} we will have {1,1,2,2} and for {3,3,27} we will have {1,1,3}. The overall type is then {{1,1,2,2},{1,1,3}. This perhaps demonstrates that multisets of signatures will play a key role in understanding multiset systems. The only other detail is the multiplicity of the empty set, in the not necessarily nullfree case, which is simply a non-negative integer. These multisets of signatures can also be acquired from the membership signatures of each dimember of a multiset system, which generalizes signatures of set systems themselves.

A larger example is {2,2,4,3,3,9,5,125,125,3125,3125,3125,7,343} which produces the multiset system type {{1,1,2},{1,1,2},{1,3,5},{1,3}}. This demonstrates that this multiset of signatures can have repetition as we can see that {2,2,4} is isomorphic to {3,3,9} as a multiset system of prime factorizations. Together this fully defines these multiset systems that emerge from commutative groups. The height of each element of each total order is the support size of each of these multisets, so we can get the order type as well from this full description.

Commutative aperiodic semigroups as multiset systems

The classification of finite commutative groups is relatively straightforward and has essentially already been achieved in terms of certain prime factorizations (which are of course multisets themselves). The harder problem then is to consider commutative aperiodic semigroups, so most of the work has went into examining them in detail. A brief description of some of these commutative aperiodic semigroups is therefore described below. All commutative associative operations are operations on multisets. Semilattices on the other hand are like operations on sets, and their characteristic operation is the closure of certain sets in a set system.

In order to generalize that to commutative aperiodic semigroups we need to define semigroups that can be described by their closure operations. To do this first we determine the idempotents by the non-repeating elements in the multisets of the set system, for these the product is equal to themselves. Otherwise the closure operation is the typical closure operation in the distributive lattice of multisets corresponding to the smallest parent. In this way, we can generalize the notion of a semilattice. There is one exceptional case size four, and then there are eighteen more of size five, and the exceptions increase within the class of aperiodic semigroups.

Size zero:

The only semigroup on zero elements corresponds to the empty set system {}.

Size one:

The only semigroup on one element is the trivial semigroup {{}}.

Size two:

There are two size two commutative aperiodic semigroups which are associated with the two types of multiset system: a set system and a multiset system. The monogenic aperiodic semigroup is clearly the multiset system because it is not idempotent and therefore it is not a semilattice.

• The semilattice: {{} {0}}
• The monogenic aperiodic semigroup : {{0} {0 0}}

Size three:

There are seven commutative aperiodic semigroups that have three elements. Since there is so few of these, it is still possible for us to give each of them different names. Since the tree semilattice is not unital, it does not include the identity element. The order of the multiset system is essentially the number of generators used to represent it.

• The total order semilattice: {{} {0} {0 1}}
• The tree semilattice: {{0} {1} {0 1}}
• The monogenic monoid: {{} {0} {0 0}}
• The monogenic semigroup plus zero: {{0} {0 0} {0 0 1}}
• The pseudozero semigroup : {{0} {1} {0 0 1}}
• The zero semigroup: {{0} {1} {0 1}}
• The monogenic semigroup: {{0} {0 0} {0 0 0}}

Size four:

Semilattices:

total order

[ [ 1, 1, 1, 1 ], 
  [ 1, 2, 2, 2 ], 
  [ 1, 2, 3, 3 ], 
  [ 1, 2, 3, 4 ] ]
{{} {0} {0 1} {0 1 2}}

weak order [1 2 1]

[ [ 1, 1, 1, 1 ], 
  [ 1, 2, 1, 2 ], 
  [ 1, 1, 3, 3 ], 
  [ 1, 2, 3, 4 ] ]
{{} {0} {1} {0 1}}

weak order [2 1 1]

[ [ 1, 1, 1, 1 ], 
  [ 1, 2, 2, 2 ], 
  [ 1, 2, 3, 2 ], 
  [ 1, 2, 2, 4 ] ]
{{0} {1} {0 1} {0 1 2}}

weak order [3 1]

[ [ 1, 1, 1, 1 ], 
  [ 1, 2, 1, 1 ], 
  [ 1, 1, 3, 1 ], 
  [ 1, 1, 1, 4 ] ]
{{0} {1} {2} {0 1 2}}

tree semilattice

[ [ 1, 1, 1, 1 ], 
  [ 1, 2, 1, 1 ], 
  [ 1, 1, 3, 3 ], 
  [ 1, 1, 3, 4 ] ]
{{0} {1} {1 2} {0 1 2}}

Near semilattices
Monogenic monoid + zero

[ [ 1, 1, 1, 4 ], 
  [ 1, 1, 2, 4 ], 
  [ 1, 2, 3, 4 ], 
  [ 4, 4, 4, 4 ] ]
{{} {2} {2 2} {2 2 4}}

Bi-idempotent near-zero semigroup + zero

[ [ 1, 1, 1, 4 ], 
  [ 1, 1, 1, 4 ], 
  [ 1, 1, 3, 4 ], 
  [ 4, 4, 4, 4 ] ]
{{2} {2 2} {2 2 3} {2 2 3 4}}

Monogenic semigroup + 0 + 0

[ [ 1, 1, 3, 4 ], 
  [ 1, 1, 3, 4 ], 
  [ 3, 3, 3, 3 ], 
  [ 4, 4, 3, 4 ] ]
{{2} {2 2} {2 2 4} {2 2 3 4}}

2-monogenic and 1-monogenic semigroups linked by a new zero element

[ [ 1, 1, 3, 3 ], 
  [ 1, 1, 3, 3 ], 
  [ 3, 3, 3, 3 ], 
  [ 3, 3, 3, 4 ] ]
{{4} {2} {2 2} {2 2 4 3}}

Monogenic monoid + identity

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 2, 2 ], 
  [ 1, 2, 3, 3 ],
  [ 1, 2, 3, 4 ] ]
{{} {3} {2 3} {2 2 3}}

Bi-idempotent near-zero semigroup + identity

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ],
  [ 1, 1, 3, 3 ],
  [ 1, 2, 3, 4 ] ]
{{} {3} {2} {2 2 3}}

Exceptional tri-idempotent semigroup

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ], 
  [ 1, 1, 3, 1 ], 
  [ 1, 2, 1, 4 ] ]
{{3} {4} {2 4} {2 2 3 4}}

Blockwise near-zero semigroup with total order

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 3, 3 ], 
  [ 1, 1, 3, 4 ] ]
{{4} {3 4} {2} {2 2 3 4}} 

Tri-idempotent near-zero semigroup

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 3, 1 ], 
  [ 1, 1, 1, 4 ] ]
{{2} {3} {4} {2 2 3 4}}

Max index two semigroups:
Zero semigroup + new zero (4 4 4 4)

[ [ 1, 1, 1, 4 ], 
  [ 1, 1, 1, 4 ], 
  [ 1, 1, 1, 4 ], 
  [ 4, 4, 4, 4 ] ]
{{2} {3} {2 2 3 3} {2 2 3 3 4}

Zero semigroup + identity (1 2 3 4)

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ], 
  [ 1, 1, 1, 3 ], 
  [ 1, 2, 3, 4 ] ]

{{} {2} {3} {2 2 3 3}}
Zero-semigroup plus a new idempotent element (1 2 2 4)

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ],
  [ 1, 1, 1, 2 ], 
  [ 1, 2, 2, 4 ] ]
{{3} {4} {3 4} {3 3 4}}

Zero-semigroup plus a new idempotent element (1 1 3 4)

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 3 ], 
  [ 1, 1, 3, 4 ] ]
{{3} {3 4} {2} {2 2 3 3 4}}

Bi-idempotent near-zero semigroup (1 1 1 4)

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 4 ] ]
{{4} {2} {3} {2 2 3 3 4}}

Non-unique iteration

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 2, 2 ], 
  [ 1, 2, 3, 3 ], 
  [ 1, 2, 3, 3 ] ]
{{4} {4 4} {2 4 4} {2 2 4 4}}

One element nilpotent the other is notand otherwise the elements go to zero

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 3, 3 ], 
  [ 1, 1, 3, 3 ] ]
{{4} {4 4} {2} {2 2 4 4}}

Semigroups of max index three
Monogenic semigroup of index 3 with identity element

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ], 
  [ 1, 1, 2, 3 ], 
  [ 1, 2, 3, 4 ] ]
{{} {0} {0 0} {0 0 0}}

Monogenic semigroup of index 3 with a new zero element

[ [ 1, 1, 1, 4 ], 
  [ 1, 1, 1, 4 ], 
  [ 1, 1, 2, 4 ], 
  [ 4, 4, 4, 4 ] ]
{{3} {3 3} {3 3 3} {3 3 3 4}}

Monogenic semigroup of index 3 with lesser idempotent

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 2, 1 ], 
  [ 1, 1, 1, 4 ] ]
{{4} {3} {3 3} {3 3 3 4}}

Monogenic with semigroup with element that is index two but for which 3,4 is not directly nilpotent

 
[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ], 
  [ 1, 1, 2, 2 ] ]
{{4} {3} {3 4 4} {3 3 4 4 4}}

Monogenic semigroup plus a zero

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 2 ] ]
{{4} {3} {4 4} {3 3 4 4 4}}

Monogenic semigroup with lesser element that is nilpotent but for which the generator and the new element do not directly go to zero Reducibles: 1,2

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 2, 2 ], 
  [ 1, 1, 2, 2 ] ]
{{3} {4} {3 3 4 4} {3 3 3 4 4 4}}

The aperiodic monogenic semigroup:
Aperiodic monogenic semigroup:

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 3 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 3, 1, 2 ] ]
{{0} {0 0} {0 0 0} {0 0 0}}

Finite commutative groups and multiplicative partitions

There are two fundamental types of semigroups: groups and the group-free semigroups which we call aperiodic (Krohn-Rhodes theory decomposes semigroups in terms of these). In particular, these are the two most basic types of commutative semigroups. The theory of finite commutative groups is completely solved, by their fundamental theorem. Finite commutative aperiodic semigroups have a more interesting theory, so they need to be considered more later. Nonetheless, the fact that the isomorphism types of finite commutative aperiodic semigroups correspond to multiplicative partitions is worth considering.

{}
{2}
{3}
{4} 
{2,2}
{5}
{2,3}
{7}
{2,2,2}
{2,4}
{8}
{3,3}
{9}
{2,5}
{11}
{2,2,3}
{4,3}

All multiplicative partitions correspond to multisets of multisets, which further can be classified into four types based upon the distinctiveness of their members as previously mentioned. Finite order-rank set systems correspond to a special type of multiplicative partition. An alternative is to classify by the equality of members rather then their distictiveness.

• Multiclan: a multiset of multisets
• Equal multiclan: a multiset of multisets such that each multiset is equal to one another
• Max order one multiclan: a multiset of equal multisets
• Equal max order one multiclan: a multiset of equal multisets all of which are equal to one another

The prime power multiplicative partitions of a number correspond to nullfree max order one multiclans, so the interesting thing is that these finite abelian groups have a multiset systems theory associated with them.

Theorem. a prime power multiplicative partition forms a disjoint union of total preorders under divisibility.

Proof. let $S$ be the support of the product of the elements of the prime power partition, in other words, all the distinct prime factors of all the elements of the multiplicative partition. Then for each element of $x$ of $S$ we can get the supermembers of the singleton set ${x}$ of the multiset of prime factorization multisets of the system, or in other words, all the numbers divided by that prime number $x$. Then by the fact that this a prime power multiplicative partition, we know that these are all disjoint from one another, as no number contains two distinct primes as divisors. Then in order to see that these are total preorders, we will prove that the support of these elements are total orders. A set of distinct powers of a prime is totally ordered under divisibility because the set of distinct prime powers can be embedded in the total order of the natural numbers by mapping the exponent to a natural number and then using that to determine divisibility. Therefore the whole collection is a disjoint union of total preorders.

A basic example is the distinct multiplicative partition of 216 which is {2,4,3,9} which being a distinct partition forms a disjoint union of total orders rather then preorders. This is equal to the partial order 2+2 formed by combining the two total orders with two elements. This is just a small tidbit about these commutative groups. Next we will further explore finite commutative aperiodic semigroups.

Types of multiset systems

A set of sets is often called a family. Therefore, a multiset of sets can be called a multifamily. The idea of a set of multisets has rarely been considered, owing to the rarity of studies of multisets. There are a couple of publications that call them "macrosets" but that is a bit verbose and then adding multi to that wouldn't be right. I developed the idea of calling them clans instead, so we could have a better understanding of the theory.

• Family: a set of sets
• Multifamily: a multisets of sets
• Clan: a set of multisets
• Multiclan: a multiset of multisets

As I discussed earlier the fundamental nature of sets and multisets emerges from commutative operations and particularly the fact that the simplest arithmetic operations like addition and multiplication are commutative. In particular, multisets directly emerge from the prime factorizations of any number. Then prime distributions are the relative frequencies of the factorization, prime signatures are the signatures of these multisets, etc. All the different types of multiset systems have corresponding types of multiplicative partitions in number theory.

• Families: sets of squarefree numbers
• Multifamily: multisets of squarefree numbers
• Clan: any set of numbers
• Multifamily: any multiset of numbers

So for example {2,6,30} is a set of square free numbers which corresponds to a set system, actually a progression family. Well {2,2,3,3} corresponds to a unary multifamily because it is a multiset of prime numbers. Then {1,2,3,4} is a set of numbers well anything like {12,12} is going to be none of these types of structures. Here is an interesting tidbit {2,6} is a set of square free numbers which corresponds to a kuratowski pair set system well their product 12 corresponds to a kuratowski pair multiset because multiplication corresponds to multiset addition due to the freeness of the multiplication operation.

References:
Toward a Formal Macroset Theory:
https://dl.acm.org/citation.cfm?id=721844

The exceptional small commutative aperiodic semigroup

Commutative aperiodic semigroups can be studied in order to consider generalizations of semilattices, and to determine their lattice like properties. Ultimately, we can define order-preserving commutative aperiodic semigroups by the joins of distinct irreducible elements are preserved under the algebraic preordering. It is clear that this is the case for all commutative aperiodic semigroups of size three or less, but in the case of aperiodic semigroups of size four it is true for 30/31 of them, all except for a single exceptional case. This semigroup is identified as the GAP small semigroup 4, 34 and it can be considered the simplest counter-case to the order-preserving property.

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 2, 1 ], 
  [ 1, 1, 1, 2 ] ]

Every single semigroup is associated with a preordering relation, and for commutative aperiodic semigroups this is a partial ordering relation. In the case of this commutative aperiodic semigroup, its algebraic ordering is the weak order [2 1 1], which is shown below.



The main issue with this small commutative aperiodic semigroup is that the product of 3 and 4 goes to 1 rather then to 2 which is the least upper bound, or the join, of the algebraic preordering. As a result, it does not preserve the ordering of its elements in its operation. It is clear that this is the unique smallest commutative aperiodic semigroup that has this property because it must have the algebraic preordering [2 1 1] so that distinct minimal elements can go to an upper bound which is different from the least one and this is the only such semigroup that does. This principle allows us to compare commutative aperiodic semigroups to semilattices in order to better generalize the properties of lattice theory to other commutative structures.

Generalizing semilattices

The generalizations of semilattices should be commutative and aperiodic (or group-free) because these semigroups are most similar to semilattices. Together, these two conditions ensure that every finite semigroup will have an algebraic preorder that is antisymmetric, that is that the elements of the semigroup are partially ordered. This isn't true for non-aperiodic semigroups, for example, abelian groups are obviously symmetrically ordered as every element can be acquired from every other one due to reversibility. It also isn't true for even the simplest non-commutative semigroup which is also a band as well as aperiodic.

A brief examination of commutative aperiodic semigroups reveals that not all of them tend tend to have the most semilattice-like properties. The first aperiodic commutative semigroup that I noticed did not look like a semilattice is the GAP small semigroup 4, 34 which doesn't preserve order joins over distinct elements of its minimal generating set. It is weak ordered by [2 1 1] and the distinct minimal elements go to the maximal element, rather then the middle which is also the least upper bound with respect to that algebraic ordering. This leads to the following definition.

Definition. a commutative aperiodic semigroup is order-preserving if it has a presentation over a set of generators such that the product of two elements a*b is equal to the join (least upper bound) of the algebraic preorder of the semigroup.

This leads to a broad generalization of semilattices within the category of commutative aperiodic semigroups. Then to consider the semilatticeness of a commutative aperiodic semigroup one could consider the number of relations that need to be added to the presentation in order to define the semilattice besides the order ones which are simply assumed.

This leads to the notion of commutative aperiodic semigroups which can be determined by multiset systems, rather then by set systems as semilattices are. Semilattices are defined by their join representations, which produce join representation families in the theory of set systems, for example, finite power sets correspond to the join representations of boolean algebras.

Commutative operations

Set theory, the theory of set systems, etc provides the best foundation for mathematics. Sets are embedded within a distributive lattice so they have commutative operations of union and intersection. Set systems can therefore be considered in terms of the union operation. Noticeably, the union operation produces redundant elements so in the set system {{0 1} {2 3} {0 1 2 3}} the elements {0 1} and {2 3} can be eliminated. Likewise, in {{0 1} {0 2} {1 2}} any element can be eliminated as redundant without effecting the union.

The missing element is the free commutative monoid, because then there are no identities in the presentation which can be used to eliminate elements except for the identity element. The identity element is always a redundant element that doesn't effect an operation so there is nothing we can do about that. Thats why small set systems of a certain size avoid considerations about the inclusion of the identity element. So for {{0 1} {0 2} {1 2}} the sum in the free commutative monoid is {0 0 1 1 2 2} and the elimination of any element strictly reduces the result of the operation. So the most interesting element is free commutative monoids which of course produce multisets.

Sets are directly linked to semilattices because they forbid repetition, and semilattices make any repetition redundant to the result of the operation. This is why there is an underlying set (the support) of any multiset produced by the idempotent-reduction of the multiset in the free commutative monoid. A natural question is how to generalize semilattices. The first that comes to mind in generalizing semilattices is that the commutative semigroup generalization should be aperiodic, because that preserves the antisymmetry of the algebraic preordering. The closest thing to a semilattice is an aperidoic commutative max index two unique non-idempotent semigroup, because semilattices require every element to be idempotent and these structures only have one element that is not. The max index of the aperidoic commutative semigroup is the max multiplicity of irredundant representations.

Many more aperiodic commutative semigroups share properties with semilattices though, without being the most similar to them. In particular, they can tend to preserve the monotonicity of presentations which is true until the weak ordered [2 1 1] commutative aperiodic semigroup of order four which doesn't preserve joins of its minimal elements. Commutative semigroups are clearly the closest to commutative semigroups, which is why we can compare the combination of elements to their joins to see similarity to semilattices and the semilattice decompositions feature in their decomposition theory. Its also worth mentioning max order two abelian groups like the xor operation which generalize the symmetric difference and therefore operate on sets, but they are not aperiodic and as groups their preorder is actually symmetric rather then antisymmetric which is what we wanted.



Commutative semigroups can naturally be described by lattice theory by considering presentations in terms of the distributive lattice of multisets. The questions of commutative semigroups then are related to their presentation in this lattice, and its operations like interiors and closures. In the other direction considerations of sets of multisets (as subsets of the multiset inclusion lattice) can be used to better understand commutative semigroups. In this sense, commutative semigroups relate the most to lattices and semilattices.

We considered commutative semigroups and their relation to lattices. However, there are two main objects of study in basic abstract algebra: structures with a single binary operation and structures with two binary operations related to one another. We can generalize from semilattices to abandon idempotence which gets us commutative semigroups, but something different happens when we abandon idempotence in lattices. A commutative semigroup pair (S,+,*) contains two commutative semigroups related by distributivity.

• The operations + and * are semigroups
• Commutativity : the operations are both commutative
• Distributivity : the operations distribute over one another

By this definition all distributive lattices and commutative hemirings are special cases and of course this means that commutative rings are as well as they are a special case of commutative hemirings. Commutative rings are of course a common common area of study then commutative hemirings because the relation between congruence and ideals in them. Importance is lended to commutative hemirings because of the fact that the natural numbers (N,+,*) are one of them which means all arithmetic can be described by hemiring theory. On the other hand, logical operations like boolean algebra and set theory can be described by distributive lattices. It is fascinating to notice this relation between fundamental arithmetic and logic.

The combined conditions of commutativity and distributivity mean that any multivariate polynomial can be described as a multiset of multisets, or a multiset system in some normal form. This relates the theory of multiset systems we are familiar with to the theories of commutative algebra and algebraic geometry. In boolean algebra, this is the relationship between set systems and boolean operations which plays a key role in the set theoretic ontology. This property noticeably only works for distributive lattices though, so for other lattices polynomials can take different forms. Lattice theory can be divided into two halves (1) the theory of distributive lattices and (2) the theory of non-distributive lattices. So a great many important lattices don't have the distributive property.

The commutative hemiring (N,+,*) is actually described by the free commutative monoid on one generator and by the fundamental theorem of arithmetic the free commutative monoid on an infinite generating set adjoined with a zero element. Which means multiplication is free except when dealing with the zero element, which absorbs or annihilates everything else as redundant when it appears in an expression. This means that fundamentally arithmetic is just the combination of multisets keeping repetition. Which demonstrates why free commutative monoids are the most fundamental concept we introduced when studying commutative semigroup theory.

I want the initial guiding principle of commutative hemiring theory to be the actually be the study of iteration, which can then be used in any semigroup because all monogenic semigroups are commutative (as seen in the ontology of commutative semigroups above). Addition is the composition of iterations, and multiplication is repeated iteration. In this way, modular arithmetic forms a commutative ring which describes the iteration properties of automorphisms. Eventually modular arithmetic could be used to describe the iteration of any monogenic semigroup element. Natural numbers describe iteration of functions without congruences, integers exactly describe iteration of functions and elements with inverses, and the rational and real numbers describe fractional iterations. Selecting structures with a specific purpose limits the selection of arbitrary structures.

Besides the theory of arithmetic and monogenic semigroups, it is noted that the subsemigroups of a semigroup form a lattice which means lattice theory can also be used in understanding general semigroups. Monogenic semigroups include the smallest possible semigroups because if a semigroup is not monogenic, a smaller monogenic semigroup can always be produced by determining the subsemigroup generated by a single element. Monogenic semigroups are also commutative, which means that certain commutative semigroups build up all semigroups. In this sense, commutative semigroups can be used to understand all semigroups. In finite group theory, the join irreducibles are actually precisely the cyclic groups of prime power order.

Commutative operations are strictly simpler then general binary operations, because they use less information then them. The output of a commutative operation is entirely determined by the underlying multiset. This property is also shared by left and right invariant binary operations, which only use the first or second argument to determine their output. These types of binary operations aren't that interesting though, because they are entirely determined by ordinary functions. This leaves us with the key of commutative operations.

Ternary closure operations

Given a ternary relation, the set of functional dependencies upon the slots of the relation form a closure operation that takes some set of slots and that produces the remaining slots that can be determined functionally from the values provided. The fixed points of this closure operation produce a Moore family, which is a set of sets that is intersection closed and that contains a greatest member. For ternary relations, this is an order three Moore family as the union of the sets of the moore family has cardinality three. There are 61 Moore families on three elements, and they are listed below. The dimension of the Moore families corresponds to the functional dimension of the classes of relations. In this way, we can produce a full ontology of all the classes of ternary relations determined by functional dependencies.

Dimension 0

trivial
{{0 1 2}}

Dimension 1

reversible & reversible
{{} {0 1 2}}

constant & reversible
{{0 1 2} {2}}
{{0 1 2} {1}}
{{0 1 2} {0}}

constant & constant
{{0 1 2} {1 2}}
{{0 1} {0 1 2}}
{{0 1 2} {0 2}}

constant & function
{{0 1 2} {2} {1 2}}
{{0 1} {0 1 2} {0}}
{{0 1 2} {2} {0 2}}
{{0 1 2} {0 2} {0}}
{{0 1 2} {1} {1 2}}
{{0 1} {0 1 2} {1}}

reversible binary operation which is partially invariant
{{} {0 1 2} {1}}
{{} {0 1 2} {2}}
{{} {0 1 2} {0}}

product to a reversible pair
{{} {0 1 2} {1 2}}
{{} {0 1 2} {0 2}}
{{0 1} {} {0 1 2}}

product to a functional pair
{{} {0 1 2} {2} {1 2}}
{{} {0 1 2} {0 2} {0}}
{{} {0 1 2} {2} {0 2}}
{{} {0 1 2} {1} {1 2}}
{{0 1} {} {0 1 2} {1}}
{{0 1} {} {0 1 2} {0}}

reversible binary operation
{{} {0 1 2} {1} {0}}
{{} {0 1 2} {2} {0}}
{{} {0 1 2} {2} {1}}

any product function
{{} {0 1 2} {2} {0 2} {0}}
{{0 1} {} {0 1 2} {1} {0}}
{{} {0 1 2} {2} {1} {1 2}}

Dimension 2

constant binary operation (disconnected)
{{0 1} {0 1 2} {1} {1 2}}
{{0 1 2} {2} {1 2} {0 2}}
{{0 1} {0 1 2} {0 2} {0}}

1 goes to 2 and 2 goes to 1 (disconnected)
{{} {0 1 2} {1 2} {0}}
{{} {0 1 2} {1} {0 2}}
{{0 1} {} {0 1 2} {2}}

partially invariant operation
{{} {0 1 2} {2} {1} {1 2} {0 2}}
{{} {0 1 2} {2} {1 2} {0 2} {0}}
{{0 1} {} {0 1 2} {1} {1 2} {0}}
{{0 1} {} {0 1 2} {2} {0 2} {0}}
{{0 1} {} {0 1 2} {1} {0 2} {0}}
{{0 1} {} {0 1 2} {2} {1} {1 2}}

partially reversible binary operation
{{0 1} {} {0 1 2} {2} {1}}
{{0 1} {} {0 1 2} {2} {0}}
{{} {0 1 2} {1} {0 2} {0}}
{{} {0 1 2} {2} {1 2} {0}}
{{} {0 1 2} {1} {1 2} {0}}
{{} {0 1 2} {2} {1} {0 2}}

both 1 and 2 go to 0
{{} {0 1 2} {2} {1 2} {0 2}}
{{0 1} {} {0 1 2} {1} {1 2}}
{{0 1} {} {0 1 2} {0 2} {0}}

completely cancellative binary operaiton
{{} {0 1 2} {2} {1} {0}}

partially cancellative binary operation
{{0 1} {} {0 1 2} {2} {1} {0}}
{{} {0 1 2} {2} {1} {1 2} {0}}
{{} {0 1 2} {2} {1} {0 2} {0}}

any binary operations
{{0 1} {} {0 1 2} {2} {1} {1 2} {0}}
{{} {0 1 2} {2} {1} {1 2} {0 2} {0}}
{{0 1} {} {0 1 2} {2} {1} {0 2} {0}}

Dimension 3

any ternary relation (disconnected)
{{0 1} {} {0 1 2} {2} {1} {1 2} {0 2} {0}}

Ontology of ternary relations

Having considered the concept of functional dimension as a means of classifying the different types of n-ary relations, we can consider the particular case of ternary relations. The notion of functional dimension is monotone so it provides an ordering of different classes of relations which can be used to build one another up. The simplest case is always the trivial case of a relation which contains no more then one element, which is used to build up the other classes.

This provides a basic upper ontology whose elements form a chain due to the monotone function to the total ordering. This classification is based upon the idea of the central importance of the notion of a function. When considering the notion of a function the first thing I consider is the different mapping patterns a function can take as well as the different kinds of functions between different partially ordered sets. There are functions from numbers like sequences, to numbers like distributions, to sets like multi-valued functions, as well as functions between orders like monotone functions, closure functions, etc. Besides this initial analysis though, our notions of functions can be extended in at least two ways: to product functions and to binary operations both of which are different.

Consider calculus, functions from a scalar to a vector and functions from a vector to a scalar are considerably different. Functions from a scalar to a vector, like the trajectories in kinematics equations act like product functions as different functions can be formed for each coordinate. So for example we could form separate functions like x(t), y(t), etc for the different coordinates and these could be differentiated separately using ordinary differentiation. On the other hand, a function taking multiple arguments would require partial differentiation. The sticking point is the number of arguments to the function (essentially the functional dimension).

In terms of ternary relations then we are considering the different types of relation based upon the number of arguments needed to express them as functions. The trivial case involves functions that take no arguments f : () -> (a,b,c) which are relations which have no more then one argument, then there are product functions f : (a) -> (b,c) which are defined by the product of two functions $f_1$ and $f_2$ which produce the first and second values of the function, then there are binary operations f : (a,b) -> c which take two arguments and which you are probably familiar with. Lastly there are all the other ternary relations.

Trivial functions:
Trivial functions f: () -> (a,b,c) have no more then one argument so they include {}, {(0,0,0)}, {(0,0,1)}, {(0,1,0)},{(1,0,0)},{(0,1,2)}, and other ternary relations isomorphic to them. There are only six possible isomorphism types in this class so it is absolutely trivial, but it is used to build up all the other ternary relations since its the simplest and smallest case.

Product functions:
Product functions f: (a) -> (b,c) are any relation which can be expressed as the product of two functions. A special case is a reversible binary operation f: (a) -> (b,c) which is also a binary operation (b,c) -> (a). These are one to one functions from some set to pairs. They reversibly deconstruct some data structure, like the reversible deconstruction of a multiset into its relative multiplicities distribution and its exponent. Reversible binary operations are of considerable interest in the theory of reversible functions. Another special case is product functions in which one of the two functions used in the product is reversible. If both of them are reversible then the relation is actually one to one to one. This is by generalization of the notion of one to one to ternary relations, so each slot determines all the others.

Binary operations:
Binary operations f : (a,b) -> (c) are an even more general class, which includes the product functions as well as numerous other operations. We already mentioned reversible binary operations, which are actually product functions single a value produces the others to reverse the operation. So the main types of operation of interest to us here are ones that cannot already be described by product functions. Probably the most important properties of operations are closure or being a magma and associativity. Both properties are used to define semigroups, which are especially important because they describe the composition of a collection of functions which transform some set. Numerous classes of semigroups like monoids, groups, commutative semigroups, semilattices, unital semilattices, abelian groups, etc make up the upper ontology of binary operations which is too extensive to list here.

General relations:
As we move to general classes of ternary relations I would like to move to closure operations on the class of ternary relations, which are classes associated with functions we can use to define larger ternary relations. Relations without any particular functional dependencies have no limit to how big they can get, so they can always be made bigger with closure operations. The first thing to notice is the different types of symmetry for ternary relations, like front symmetry, back symmetry, and outer symmetry which is similar to symmetry for binary relations but for projections of the ternary relations. Then there are cyclic closed ternary relations which are relations for which each rotation of a member is included. Cyclic orders are of this class well betweenness relations are outer symmetric.

Functional dimension

A n-ary relation is a set of sequences that each have n slots with values associated with them. Functional dependencies can be used to describe how different slots in relation are dependent upon one another, so for example in a binary relation we can describe that the second value is dependent upon the first, which is the typical definition of a unary operation. This produces a closure operation on the set of slots, so that given some set of slots, the closure can be used to produce the information which can be inferred from them.

Closure operations are a common part of mathematics, and they are associated with lattices and moore families. For example, given a group the subgroup generated by a set is a closure operation, and in particular in linear algebra the span generated by a set of vectors is a closure operation. Also in linear algebra, the dimension is the smallest number of generators needed to span the entire vector space. It is for this reason that I call the functional dimension the number of slots needed to generate all the elements of a relation.

The interesting thing is that this is a monotone function function like cardinality, so the subrelations of a given n-ary relation must have less functional dimension then their parents. This monotone function can be used to help to classify relations and their ordering.

Prime distributions

Number theory is a great source of multisets, perhaps a better source then set systems as not all finite multisets are used there. Numerous number theoretic functions correspond to multiset operations applied to the prime factorization multiset.

• $\omega(n)$ the order of the prime factorization multiset
• $\Omega(n)$ the cardinality of the prime factorization multiset
• d(n) the number of parts of the multiset
• exponent(n) the exponent of the prime factorization multiset
• prime-signature(n) the signature of the prime factorization multiset

Given any set we can produce a multiset from that set by applying a function to each of its members. If this function is irreversible, this produces a multiset whose partition is more coarse then the set that proceeded it, this is why I call this functional coarsification. This can then be transformed to a relative frequency distribution. This allows us to get a relative frequency distribution for the application of any irreversible function to a set, which is an abundant source of relative frequency distributions. For example, we can apply this to the $\omega$ function for the range from one to a thousand.

{0 0.002, 
 1 0.193, 
 2 0.507, 
 3 0.275, 
 4 0.023}

This can then be compared to the application of $\omega$ from one to a hundred thousand. One see that here the distribution tends towards three to a greater extent.

{0 2.0E-5, 
 1 0.097, 
 2 0.33758, 
 3 0.38844, 
 4 0.15855, 
 5 0.01816, 
 6 2.5E-4}

It should be readily apparent that larger numbers tend to have more prime factors. This is because the divisor ordering is a suborder of the total ordering on the natural numbers. This describes the distribution of the $\omega$ function over small ranges but the Erdos-Kac theorem of probabilistic number theory says that omega(n) is normally distributed with variance and mean $log(log(n))$. This is sometimes expressed as a Poisson distribution rather then a normal distribution because the mean and variance are both the same. This describes the rate that the number of prime factors grows at on average. Probabilistic number theory is the perfect mix of multisets and probabilities, and it demonstrates how the two are intertwined as expressed through the positive integers themselves.

In a complete set of multisets (one that is submultiset closed and finite union closed) different atoms appear as independent random variables, this means that in the set of positive integers the primes act like independent random variables. This is because the positive integers are in some sense complete, as all prime factorizations appear. If they didn't this fact from probabilistic number theory certainly wouldn't work.

Noticing that positive integers themselves are multisets we can produce relative frequency distributions from them which describe the extent to which each prime is distributed within the number. For example, given the number twelve we can produce the distribution {2 2/3, 3 1/3} which of course would be the same for 144. This is associated with the values {1/3 2/3} which describe the relative frequencies of the distribution. These frequencies add up to one, so ontologically this is called a unit partition. To explore probabilistic number theory more I described the different unit partition types of different numbers within a range and how these in turn are distributed. Here are the unit fraction partitions of the first one hundred numbers and how they are themselves distributed within that range.

{*{1/3 1/3 1/3} 0.05,
 *{2/5 3/5} 0.01, 
 *{} 0.01, 
 *{1/5 4/5} 0.02, 
 *{1/2 1/2} 0.32, 
 *{1/4 1/4 1/2} 0.03, 
 *{1/3 2/3} 0.15, 
 *{1/4 3/4} 0.05, 
 *{1/6 5/6} 0.01, 
 *{1} 0.35}

A regular number, that is to say a number with a regular multiset for its prime factorization, is also a square-free power. As we can see here the main unit fraction types described at this range are the regular distributions of order one or two. The unit fraction distribution produces a finer partition then the omega function, so the different partitions are distributed in a way corresponding to the number of distinct prime factors. An antiregular number will have a unit partition that is a set rather then simply a multiset. Rather the number is a squarefree power, antiregular, or of a certain order is determined by the prime distribution of the number. Here are some more examples of prime distributions.

• The prime distribution of 360 is {3 1/3, 2 1/2, 5 1/6} this is antiregular.
• The prime distribution of 384 is {3 1/8, 2 7/8} which is highly irregular, as one prime is far more frequent then the other.
• The prime distribution of 900 is {2 1/3, 3 1/3, 5 1/3} which is regular because this is a squarefree power.
• The prime distribution of 1260 is {7 1/6, 3 1/3, 2 1/3, 5 1/6}.

We talked about the unit partitions and the $\omega$ function dealing with the distinct primes, but the different prime distributions themselves are distributed within the natural numbers in a certain fashion. This isn't too hard to find out, since each prime distribution can be expressed as an exponential equation $b^n$ for some base $b$ which has coprime multiplicities of prime factors, or in other words which is of exponent one. The counting function is therefore logarithmic, and the probability can be computed relatively from that. \[ \frac{log(n)}{log(b)*n} \] This grows asymptotically at degree negative one, which means that individual prime distributions rarely occur (as can also be inferred by their logarithmic counting function). There are an infinite number of prime distributions distributed within the positive integers, but each of them occurring relatively rarely but infinitely nonetheless.

Notice that the natural numbers can be represented as multisets of a single element, which means that the total ordering of natural numbers is simply the inclusion relation on them, then the divisibility relation is simply the relation of even divisibility of these multisets. Next, we can introduce the concept of even disibility of the prime factorization multiset instead. Another way of expressing this is that there exists a positive integer $n$ such that $a^n = b$. In this partial order, the connected components are the numbers with the same prime distributions, as only numbers with the same prime distributions can be exponentiated to get one another. This only occurs when the exponent of two numbers with the same prime distribution divide one another. This generalizes divisibilty ordering associated with multiplication to an ordering associated with exponentation. Each of these partial orders is a suborder of one another.

Relative frequency distributions of multisets

Let $A$ be a multiset. Then we can add $A$ to itself repeatedly to get $A+A$, $A+A+A$, and so on. The multisets {x,x,y},{x,x,x,x,y,y}, and {x,x,x,x,x,x,y,y,y} for example are all produced by repeatedly adding some multiset together. This means that they are equivalent to scale. The multisets {a,b,c,d,d} and {a,a,b,b,c,c,d,d,d,d} are also equivalent to scale for the same reason. This produces an equivalence relation of multisets. We can use the concept of relative frequencies to characterize this equivalence relation of multisets.

Definition: the relative frequency of some element of the support of a multiset is equal to $m/n$ where $m$ is equal to the multiplicity of the element and $n$ is equal to the cardinality of the multiset. The relative frequency distribution is the support set of the multiset with each element with each element weighted by its relative frequencies.

Definition: the exponent of a multiset is equal to the greatest common divisor of its set of multiplicities.

Using the concept of relative frequencies so defined, we can see that the relative frequency distribution of the first class of multisets is {x 2/3, y 1/3} and the relative frequency distribution of the second class of multisets is {a 1/5, b 1/5, c 1/5, d 2/5}. Multisets with a given distribution can be scaled in an infinite number of ways to get different multisets. These different multisets all belong to the same equivalence class. The multiset exponent describes the scale of the multiset.

Theorem: the product function from a multiset to the pair (relative frequency distribution, exponent) is reversible.

Proof: suppose that we have a pair (relative frequency distribution, exponent) then we can construct the corresponding multiset by scaling each relative frequency of the multiset by $lcm(D)*n$ where $D$ is the set of denominators of the relative frequency distribution and $n$ is the exponent. To see this consider first what happens when we scale the relative frequency distribution by $lcm(D)$. What we want to show is that this produces a coprime multiset, that is a multiset with exponent one.

Firstly, in order for the resulting set of multiplicities to be integers they must be multiplied by a common divisor of the set of denominators of the relative frequency distribution. If they are not then there is some element in the denominator that will not be multiplied by a value that is a multiple of it, so the resulting multiset will not consist of integers. But the least common multiple (LCM), is the smallest common multiple of the integers in the divisibility lattice which means the gcd of the resulting multiplicities will be one because if it wasn't then it wouldn't be the least common multiple but rather a greater common multiple. To then get the proper multiset scale that further by the multiset exponent in the second part of the pair so that the gcd is $n$.

This product function isn't only one to one it is a complete destructuring of multisets because the equivalence class of multisets up to scale and the equivalence class of multisets with the same exponent are both independent of one another in the lattice of equivalence relations related by coarsification and refinement. This makes it especially valuable type of reversible function.

Postulate: the regularity, antiregularity, and the order of a multiset are determined by its relative frequency distribution

Actually, these are all determined by the support partition of the multiset which is the partition of the support of the multiset by equivalent multiplicites. Two relative frequencies of a given multiset are only equal if their corresponding multiplicities are because $m_1/n = m_2/n$ can be simplified by multiplying both sides by $n$ to get $m_1=m_2$ so all the properties of the support partition are preserved by the relative frequency distribution. Regular multisets produce regular distributions and antiregular multisets produce antiregular distributions.

Classification of set systems by degree multisets

In set theory in a multiset theoretic context the key role of multisets in the theory of set systems in set theory was explained. In particular, a multiset can be formed from a set system by adding all the sets of the set system together counting repetition, which determines the degree distribution of the set system. This is a brief overall survey of the different properties of set systems determined by the degree multiset.

Cardinality sum
The cardinality sum of the set system is equal to the sum of the cardinalities of the sets of the set system, and it corresponds to the cardinality of the multiset. The cardinality sum is the best measure of the size of a set system. Cardinality one multisets include {x}, cardinality two ones include {x,x},{x,y}, and so on. Cardinality sum one set systems are rank one chain families, cardinality sum two set systems are max order two independent families, and cardinality three set systems are either max order three set systems or they are ordered pair like set systems such as the kuratowski pair {{0},{0,1}}. A full survey is presented in set systems of small size .

Order
The order of a set system is the cardinality of its union, and the order of a multiset is the cardinality of its support. Set systems with bounded order have a bounded cardinality sum up to $2^n$ where $n$ is the order, and set systems with bounded cardinality sum are bounded by the same order. This means there are more set systems of a given order, so the kuratowski pair and ordered pairs are order two and so is the larger cardinality sum set system {{0},{1},{0,1}} which is power set like.

Regular families
Regular families are families for which the degree of each element is the same, and they correspond to regular multisets like {a,b,c},{a,a,b,b,c,c}, {a,a,a,b,b,b,c,c,c}, {a,a,a,a,b,b,b,b,c,c,c,c,d,d,d,d}. Corresponding regular families include {{a},{b},{c}}, {{a,b},{a,c},{b,c}}, {{a,b},{a,c},{b,c},{a,b,c}}, and {{a},{b},{c},{d},{a,b},{a,c},{b,c},{a,b,c}}. Finite power set like set systems are a special case of regular families determined by the condition that the cardinality sum is $2^n$ which is the highest it can be for the order of the set system.

Antiregular families
Antiregular families are families for which the degree of each element is different, and they correspond to antiregular multisets like {a},{a,b,b},{a,b,b,c,c,c}, and {a,b,b,c,c,c,d,d,d,d}. Actually, the antiregular multisets must all take this form as multisets like {a,b,b,b} have no corresponding set system partition so they are actually called progression multisets as their multiplicities form a range over the natural numbers. Ordered pairs can be determined by their degrees multiset which is the basis of the multiset theoretic definition of the ordered pair.

Maximum and minimum degrees
Independent families are set systems for which the intersection of all sets is empty, they are also determined by their degree multisets, as their degree multisets are actually sets. Consider the independent family {{0,1},{2,3}} its combinational sum is {0,1,2,3} which is a set, and in this case multiset addition is equal to the union. Max degree two, max degree three, and max degree four families are generalizations. Powerful families are set systems for which each union member has degree at least two. These properties are determined from the minimum and maximum multiplicity of the degrees multiset.

Repetitiveness and pseudo-independent families
The repetitiveness of a multiset is the extent to which it is not a set, or the cardinality of the multiset minus the cardinality of the support. This means that for a set system, it is equal to the cardinality sum minus the order. It is maximized for power sets and minimized for independent families. It is also equal to the total intersection size of the set system, which is the total number of times sets intersect. Every set system has a fundamental intersection multigraph associated with it, which is a multigraph rather then an ordinary graph because sets can have more then one value in common between them.

Antiregular families

Set systems can be classified by their degree multisets. Regular families are the set systems in which the degree of each union member is equal, antiregular families by contrast are the set systems for which the degree of every union member is different. Progression families like {{0} {0 1} {0 1 2} {0 1 2 3}} are antiregular because the degree of each union member is different from the others. Kuratowski pairs, rather they are equal like {{0}} or distinct like {{0} {0 1}} are also antiregular.



Antiregular families characterize the total orderings on a set (all preorders and graphs can be described as set systems), but not the total preorderings because preorders can have elements that are equal and indistinguishable. The total ordering on the set can be recovered from the multiset alone by the ordering of the multiplicities of the multiset, which is the basis of the multiset theoretic definition of the ordered pair. The corresponding progression family then be recovered from the multiset by the total ordering, which means antiregular families a special case of family that can be recovered directly from their multisets.

The same properties don't occur for sets of multisets, as sets of multisets can take a variety of forms besides progressions for example like {{0}, {1,1}}. Progressions of multisets which can be used to characterize sequences in pure multiset theory aren't always antiregular like with {{0},{0,1},{0,1,1}} which is equal to (0 1 1). This is actually a regular progression of multisets rather then an antiregular one which shows that this property of recoverability from the multiset does not apply sets of multisets like it does to sets of sets.

Regular families

Regular families are families for which every member of the union of the sets of the family has the same degree. They are a special case of the classification of set systems by their degree multisets (the ontology of multisets determines the classification by degrees). So a regular multiset by comparison is a multiset in which the multiplicity of each element is equal for example {a,a,a,b,b,b,c,c,c} is regular because the multiplicity of each element is equal. The ontology of regular families is included below.



Special cases include the rank complete families, which are set systems determined by the collection of all subsets of some set, typically produced with the selections function and the union of such sets. Another special case is the case of independent families (families whose degree multisets are actually sets which works because sets are regular multisets) which is a different classification by degrees. Similar concepts of regular families emerge from the set theoretic representation of the graphs (either maximal cliques families or dependency families).

Approaches to artificial intelligence

The good old fashioned approach to artificial intelligence (AI) is based ontology engineering and knowledge base construction. The earliest ontology languages like KIF were dialects dialects of Lisp because of its association with the artificial intelligence community. Ontology languages used today are now more often based upon XML like OWL. Given some domain, all the semantic relations between objects in that domain can be specified to create a semantic network.

The main alternative trend in artificial intelligence right now is to use neural networks. Neural networks have gained a lot more traction under the name deep learning. It is well known that neural networks have surpassed humans at go playing. Combinatorial game theory is essentially a solved problem as humans are generally going to have a hard time against a well trained neural network.

I have great optimism in the future of semantic networks, the modeling of semantic relationships between phenomena can still be greatly improved. In particular, semantic networks are especially useful for modeling symbolic thought. I think even as neural networks are improved semantic networks and ontology engineering are still worth pursuing and can still be improved on their own. Perhaps superhuman artificial general intelligence will be created with some appropriate combination of semantic networks, neural networks, and other methods.

See also: Symbolic vs Connectionist A.I.

Mathematical theories of semantic relations

All of the different semantic relations which make up an ontology have certain mathematical theories associated with them. In summary, the mathematical theories of equivalence relations, partial orders, and boolean algebras and set systems can be used to better understand certain semantic relations. This is why these concepts play a fundamental role, the theory of synonymy is the most important of all of these because it introduces the fundamental idea of the ontology of equivalence relations.

• Synonmy
• Meronomy
• Taxonomy

Mathematical theories of taxonomy and hyponymy

Set theory is the mathematical context in which relations of taxonomy and hyponymy are defined. Set theory defines a hierarchy of concepts of sets, set systems, sets of sets of sets, etc. Sets and proper classes are taxonomically related to one another if they are subsets or subclasses of one another. So taxonomy can be described by the mathematical theory of set inclusion.

Hyponomy is distinguished from taxonomy by the fact that it describes instances rather then subclasses. So the instances of hyponymy relations need not be classes themselves. Hyponomy is particularly interesting when it is applied to sets and classes. To give an example, the class of antisymmetric binary relations is a taxonomical subclass of the class of binary relations and it is an hyponym of the class of subclass closed families.

A mathematical taxonomy will include all the different sets and classes of mathematical structures, elements, and expressions that any given mathematical theory will have. A mathematical taxonomy is provided by graphclasses, but otherwise there does not seem to be a general mathematical taxonomy available. The mathematical taxonomy could be enriched with hyponomy and related relations to create a general formal ontology of mathematics.

Mathematical theories of meronomy

Merenomical relations have three axioms : reflexivity, antisymmetry, and transitivity.

• Reflexive: everything is a part of itself
• Antisymmetric: nothing is a part of its parts
• Transitive: everything that is a part of something which is a part of something else is a part of that something else

This means that merenomical relations form a partial ordering. Different types of semantic relations used in knowledge representation and ontology have different types of mathematical relations associated with them, so synonymy relations are related to equivalence relations, mereonomy is related to partial orders, etc which leads to our mathematical theory of ontology.

Multiples:
A multiple is an element of a partial order whose set of proper predecessors is upper bounded. It is then a multiple of the upper bound of its set of proper predecessors, and elements can be multiples of multiples and so on. This leads to the multiset theoretic representation of the elements of a partial order.

Join irreducibles
It can easily be seen that the join irreducible elements of a finite lattice are precisely the multiples of other elements, with atoms being the elements that cover the lower bound. A multiatomistic lattice is one whose join irreducibles are either atoms or multiples of them. Multiatomistic distributive lattices describe multiset inclusion. In the infinite case, the relation between join irreducibles and multiples of elements still holds, but it doesn't in the case that joins are restricted to pairs of elements. In that case it is only true when the elements of the lattice are lower isolated.

Suprema irreducibles
Elements of a partial order are called suprema irreducible if they are not included in the suprema of their set of proper predecessors. This concept doesn't seem to appear in the literature, so I introduced it myself as a generalization of the join irreducible elements of a lattice. Well elements of lattices can be represented by multisets (or simply sets in the case in which the lattice is atomistic) the case of general partial orders is somewhat different. Two elements of a general partial order may have the same set of suprema irreducible predecessors. The set of suprema irreducible predecessors forms a monotone function which may produce the same output for different values.

The special case of ordered collections motivates the definition of suprema irreducibles. Consider the set of ordered collections partially ordered by inclusion. In this well known partial order, the only suprema irreducibles are the equal lists. These suprema irreducibles are actually all multiatomic elements of the partial order, because they are all multiples of the singular lists. This makes the partial ordering on ordered collections multiatomistic. Emerging from this relation is the underlying multiset of any ordered collection which describes the suprema irreducible predecessors of the collection. Two elements may have the same underlying multiset, so this function loses information but it is still monotone. Since this function information these elements can be seen to have additional structure about them (structural multisets) which together with the underlying multiset describes the structure.

Structural multisets
These leads to the theory of structural multisets, which are general data structures that have an underlying multiset as well as other sets and multisets about them which describe their structure. Included within the merenomic relation of structural multisets is the notion of an induced structure, which is a structure induced by taking a submultiset of the underlying multiset of a structure. Induced substructural orderings are not necessarily lattices, and can be used to describe general partial ordering relations. The partial ordering on ordered collections is an induced substructural ordering on its multiset-theoretic suprema irreducible representations. Well ordered collections are represented as structural multisets, other structures like rings can be represented as structured sets. All composite structures can be described by structural multisets.

Mathematical theories of synonmy and equivalence

I will begin the analysis of semantic relations with the relations of synonymy and equivalence because this is an area of study that is significantly under-explored. The theory of equivalence relations is one thing that sets this ontology project apart from others like it. The theory of equivalence is often taken for granted in traditional mathematics as it starts with sets that have an equivalence relation already attached to them. The ontology of equivalence relations should be a significant part of any project in ontology with formally defined semantics.

Ontology of multisets

The difference between sets and multisets is that multisets can have repeated, or equal elements. This means that every multiset theoretically comes equipped with an equivalence relation associated with it, which tells which elements are equal to one another. Multisets can therefore be classified, ontologically, based upon the equivalence relation between elements of them. Every theory of equality should therefore start with the ontology of multisets as multisets can be classified based upon the equivalence relations on them.

As multisets have an equivalence relation on their elements, we can classify them by the extent to which their elements are equal to one another. Multiset repetitiveness is a metric which effectively describes the extent to which a multiset has elements which are equal to one another. Different classes of multisets then can be formed based upon different degrees of repetitiveness, as near sets are sets which have at most one repeated element, and sets have no repeated elements. Max multiplicity two multisets have at most one repetition for each individual element so they are a generalization of near multisets. In this way, sets emerge in the multiset ontology based upon the extent of equivalent elements in them.



Well sets are classifiable as multisets which have the least amount of equal elements there is a conjugate notion. Equal multisets have members that are all completely equal to one another, and near equal multisets have at most one element not equal to all the others, well max order two multisets have at most two distinct elements. In this way the equivalence relations on multisets can be classified either by the degree of equality of the degree of distinctiveness. The equal multisets are the most equal and sets are the most distinct. Refinement is the process of making things more distinct, well coarsification is the process of making things more equal.

Functional coarsification occurs when given some multiset, some function can be applied to each member of the multiset in a similar manner to the higher-order function map applied to lists. This then produces a new multiset that is at least as coarse, or more coarse then the previous one. Consider the power set, {{} {0} {1} {2} {0 1} {0 2} {1 2} {0 1 2}} applying the cardinality function to each member produces {0,1,1,1,2,2,2,3} which is more coarse because it is a multiset with equal elements rather then simply a set. In this way, functions as simple as cardinality can be used to produce multisets with equal members from sets. The lack of widespread support for multisets is part of the larger issue of a lack of a theory of equivalence.

Ontology of equivalence relations

The ontology of equivalence relations is not about classifying equivalence relations themselves, but rather about creating an ontology consisting of equivalence relations and the relations between them. Equivalence relations form a lattice, with operations of containment, independence, and covering produced together with them. The combination of independence and covering also produces complementation, so an ontology of equivalence relations should include knowledge about which equivalence relations are complementary to one another.

• Containment : an equivalence relation is contained in another if it is finer then it
• Independence : two equivalence relations are independent if their meet is minimal
• Covering : two equivalence relations cover one another if their join is maximal

An example of the ontology of equivalence relations that should be understand is the set of equivalence relations between numbers. Consider that certain numbers have different properties associated with them like sign and the absolute value, and these properties are part of a general process of containment. So that certain properties like absolute value, sign, etc are contained within one another. This is an ontology consisting of equivalence relations and their properties rather then a classification of equivalence relations themselves, which would be similar to the classification of multisets already described.



Equivalence relations are especially applicable to structures like lists which have different specific slots associated with them like the first element, the second element, and so on. The same is true of records and related data structures, essentially they have already established properties structurally associated with them. This leads to the ontology of equivalence relations on pairs, which are defined by the amount of data that can be acquired from the pairs. The pair of pairs relation produces no information, the equal first gets the first element, the equal second gets the second element, and the equal pairs gets the combination of both the first and second elements. These properties are combined in a partial order based upon the amount of information acquired by them.



A similar ontology of equivalence relations can be used to describe the properties of triples, quadruples, and related sorts of ordered collection structures. Multisets themselves have various properties that can be further partially ordered like signature, repetitiveness, the underlying set, order, max multiplicity, etc which leads to an ontology of equivalence relations of multisets which is the next aspect of describing multisets besides the ontology of multisets themselves. The ontology of equivalence relations should include more then just the containment relations described above, but also information about independence and covering so that rather two properties are complementary can be decided.

Ontology of binary multirelations

The ontology of equivalence relations can be used to classify multirelations. The ontology of equivalence relations of ordered pairs is especially important to the classification of binary multirelations as they consist of ordered pairs. Given a multiset and an equivalence relation, the multiset is distinct with respect to that equivalence relation if no two elements in the multiset are equal with respect to that equivalence relation.

The ontology displayed below consists of classes of binary multirelations with zero or more distinctiveness relations associated with them. Binary multirelations have no distinctiveness relations. Binary relations have one distinctiveness relation: the identity, which means that no two elements can be the same. This is the same as saying that they are essentially sets. Unary operations are further distinct with respect with respect to their first element, inverse unary operations are distinct with respect to their second element, and reversible unary operations are distinct with respect to both. Unique relations are distinct with respect to the trivial equivalence relation because they have no more then one element. All these distinctiveness relations together form the free distributive lattice on two elements.



One can immediately see that sets and functions both emerge in the same manner in the multiset theoretic ontology : from distinctiveness relations. Sets emerge from distinctiveness with respect to identity, and functions emerge from stronger notions of distinctiveness. In fact the set theoretic definition of functions is based upon the condition that each of the first elements of the relation is distinct from one another. All these relations of distinctiveness are displayed above with respect to containment.

Function and map like properties can be acquired from any collection with some kind of distinctiveness relation. The ontology of equivalence relations on numbers was described previously, so based upon that consider absolute-value distinct collections of integers. The oriented range {1,-2,3,-4,5,-6,7,-8,9} is an example of such a collection with a distinctiveness relation with respect to the absolute value. This can be considered then as a function from the range to certain signs like {1 +, 2 -, 3 +, 4 -, 5 +, 6 -, 7 +, 8 -, 9 +}. This demonstrates that functions, just like sets, are an emergent property of distinctiveness relations.

Theory of functions

The theory of functions must start from the theory of equivalence relations as every function has a kernel associated with it which states that two elements are equal when they are equal with respect to the function. This describes completely the irreversiblity of the function, as functions tend to lose information and be irreversible. The extent to which functions lose or gain information is described by the containment ordering of equivalence relations. To make a function reversible all that is needed is a complementary equivalence relation. This is how reversible destructuring operations are produced.

References

Computer-aided research of synonymy and antonymy https://www.aclweb.org/anthology/C69-5801

Semantic relations

Every ontology is based upon information concepts like classes and individuals and the semantic relations between them. In order to describe the organization of an ontology it is a good idea to take the different semantic relations between elements as a starting point. Each of the different semantic relations like synonomy, antonymy, hyponomy, merenomy, and holonymy play a fundamental role in ontology. Each of them is worth reviewing separately in more detail.

• Synonmy and antonymy : equal and opposite elements
• Hyponomy and taxonomy : generalization and specialization
• Mereonomy and holonymy : relating parts and their wholes

Generalized list processing

Lisp is commonly characterized by list processing, as lists are the fundamental data structure of Lisp. Generalizations of lists can be formed by equivalence classes to get sets, multisets, and related structures which are essentially just lists with certain amount of the ordering removed. A generalized Lisp programming language can deal with these other list-like structures such as sets and multisets (sets are even introduced in Clojure) which is especially important as sets are typically used as the foundation of mathematics. Generalized list processing deals with sets, multisets, lists, and related structures.

It can be seen that most programming in general is largely List processing, and this fact is best realized in Lisp dialects. There are some programming languages which take the associative array or some equivalent as the fundamental structure and these languages and others so commonly used do not get people to realize the truth of List processing. Lisp isn't just about homoiconity but also its choice of data structures. It just happens that it chose the type of data structure most suited for defining programs. To deal with the problem of associative arrays, I will instead reduce them to binary relations which are lists of pairs, which will be an important part of our ontology.

Equivalence classes of lists

Equivalence relations can be formed that determine that certain lists are equal to one another. One way of doing that is to take the orbit of a class of permutation groups acting on lists. These are equivalence relations that determine that lists are equal to one another up to a reordering. Different types of equivalence classes of lists are produced from symmetric groups, alternating groups, cyclic groups, and so on. Generally, transitive groups are selected here to determine collection classes.

Symmetric lists:
The first thing to realize is that symmetric lists are multisets, which means that any list can be treated as a multiset by simply ignoring its ordering. In this sense, it is perhaps not necessary to have a different data type for multisets, sets, etc (except for the purpose of creating an equivalence class). Then the important question is which operations on lists are invariant under reordering, the multiplicity function for example is invariant under reordering.

Alternating lists
Alternating lists are similar to symmetric lists, in the sense that they have relatively few permutations. Generally speaking, collections of this type are rarely mentioned.

Cyclic lists:
Cyclic lists or circular lists, are lists which are invariant up to rotation. Cyclic linked lists provide a possible implementation. Even though these are perhaps the next most useful class of lists, after examining the different types of collections determined by permutations, I concluded that symmetric lists are the most fundamental because they are a partially ordered multiset, so the most important collections are related to partially ordered multisets and aren't simply general relations.

ANSI Common Lisp

I finished most of ANSI Common Lisp. I got this one because it was made by Paul Graham who is well known in the Lisp community, and I have been trying to read as many programming books as possible. There seem to be more books on Lisp dialects then on virtual machines. Anyways, I already mentioned FSet as a possibility to add better support for sets and multisets to Common Lisp. Another possibility is to simply treat lists like sets and multisets. The Common Lisp function member is already available to you and a multiplicity function is easy to implement. One could imagine that my post on core data structures perhaps wasn't even necessary, all that is necessary is a Lisp.

Literal and symbolic ontology

The mathematical ontology can contain at least two components (1) the literal ontology which doesn't deal with symbolic elements and (2) the symbolic ontology. I already specified that the core mathematical collection types are sets, multisets, and sequences. So the literal ontology could include things like sets of sets, multisets of sets, sets of multisets, multisets of multisets, sets of sets of sets, and so on well multisets of sequences and sets of sequences are multirelations and relations respectively. Each of these different types of classes form different elements of the mathematical ontology constructed just from core data types.

The symbolic ontology can include symbols like the symbols referring to addition and multiplication, and in this way, it can deal with all the other types of data structures encountered in mathematics. All mathematical structures must be representable by some finite symbolic structure in order for them to be reasoned about. So we can classify types of polynomials like univariate polynomials, quadratic univariate polynomials, linear univariate functions, multivariable polynomials, etc. Lattice polynomials can be formed symbolically as related to lattices from abstract algebra. Eventually, even differential equations can be represented as symbolic expressions.

Mathematical equivalence relations

The only significant difference between mathematics and computation is that mathematical data structures can be implemented in multiple ways, like how a number can be a floating point number or a fraction. The same is basically true in computable set theory, as it is necessary to have multiple representations of sets. Von Neumann ordinals for example, grow exponentially so it makes more sense to represent them with a different data type with a single number rather then an exponentially large tree. Mathematics deals with abstract interfaces then rather then particular data types.

It remains to implement an equivalence relation to see if different data types have the same mathematical value. The axiom of extensionality, one of the fundamental axioms of set theory, states that two sets are equal if they have the same members regardless of their representation. It is therefore necessary to have an equivalence relation on sets that recursively check them for equality. Likewise, it is generally useful to have an equivalence relation for numbers, Clojure for example has == to check the equality of numbers and most programming languages have something similar. Mathematical meaning can be defined by an equivalence relation between computable data structures. Then mathematical ontology is the classification of objects up to the mathematical equivalence relation.