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
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.
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