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.
Thursday, November 21, 2019
Saturday, November 9, 2019
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.
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.
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