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.