The most important point of consideration during this past year was the theory of commutative operations. Commutative operations can be defined as functions of unordered pairs rather then as functions of ordered pairs. This requires the notion of a multiset which can be defined as an unordered collection of elements. Sets have the limitation that they exclude repetition, but after the generalization of multisets they can be included amongst multisets as part of an ontology of unordered collections.
The theory of commutative semigroups emerges from the study of commutative operations. One of the first things one notices is that semilattices are included among the commutative semigroups. This is how we can then get to the idea of commutative operations that generalize semilattices. These need to be aperiodic (group free) semigroups. These operations can be defined as functions that produce an upper bound of elements of a partial order that need not be the least upper bound, because upper bounding producing functions are aperiodic.
Many commutative operations of an aperiodic type actually arise directly out of semilattices from the properties of elements (such as from monotone functions). This occurs in the complement of the partition lattice for example. Partial orders are known to strictly order things from minimal to maximal, so the operations that correspond to them are aperiodic. That is to say, aperiodic operations correspond to partial orders. There are of course also commutative groups such as the cyclic group. The theory of finite commutative groups is of course well known.
Multisets can be defined as elements of the free commutative semigroup on a universal generating set. Multisets can therefore be concatenated counting repetition using this semigroup. Using the concept of multisets we can consider the nature of commutative aperiodic semigroups (especially small commutative aperiodic semigroups) using multiset systems. By analogy with the representation of semilattices as closure operations on set systems we can define certain commutative aperiodic semigroups as closure operations on multiset systems subject to certain index limitations. Of course, in the free case no closure operation is necessary and we have an operation fully defined from multiset combination.
The important concept of degree reductions produces a partial order on set systems and their generalizations based upon the idea of taking subsets of members rather then just subsets of the set. It is noticeable that the combinational multiset of combinatorial set systems is a monotone function of these degree reductions, so we can consider certain set systems as partitions of a multiset. For example, the kuratowski pair is the unique set theoretic partition of the kuratowski pair multiset. We don't even need set systems then, and we can define ordered pairs directly from multisets.
Finite multisets of course have signatures associated with them. So for example, we can see that kuratowski pair multisets have certain special signatures. These signatures can be ordered by the Young's lattice so that they are a monotone function on multisets in the same way that cardinal numbers are a monotone function on finite sets. Young's lattice therefore plays an important role in multiset theory.
There are two main ways to combine two commutative operations together (1) into a distributive lattice and (2) into a commutative ring. These commutative operations are combined by the distributive law, which ensures that polynomials do not need to be nested (hence the importance of polynomials in commutative and boolean algebra). Distributive lattices are essential in set theory and multiset theory. Both multisets and their signatures form distributive lattices. Elements of distributive lattices can be represented as sets, for example this naturally leads to the definition of von neumann ordinals. Commutative rings are used in the definition of arithmetic and they have important properties related to factorizations (which are actually multisets).
Natural arithmetic $(\mathbb{N},+,*)$ is not a commutative ring because it lacks additive inverses, but it is a commutative semiring. Both natural addition and natural multiplication are (with the single exception of multiplication by zero) free commutative operations defined by combining multisets, so the extent to which arithmetic is multiset theory is considerable. This is ensured by the fundamental theorem of arithmetic which states that every positive integer is uniquely factored into a multiset of prime numbers. Extra details are of course added on by the different types of commutative operations and inverses. But algebraic number theory is still essentially commutative.
I covered too many things in the past year to cover them all here effectively, but I mainly want to draw attention to multisets, multiset systems, commutative semigroups, commutative rings, distributive lattices, and related concepts. The main defining feature of these structurse is that like sets they lack an ordering on their elements.
