Yearly review and todo

When I was focusing on combinatorics, I was mainly concerned with set systems, hypergraphs, and related structures. As I am doing abstract algebra now, I have been increasingly concerned with computations with polynomial systems.

• Set systems (subsets of distributive lattices)
• Polynomial systems (subsets of commutative rings)

Thinking of both set systems and polynomial systems as substructures demonstrates how they are not as different as they would seem. On the one hand you have union closed and intersection closed set systems and on the other hand you have additively closed and multiplicatively closed polynomial systems. Then there are other closure systems of sets like subclass closed, superclass closed, convex set systems, etc and of rings like ideals, radical ideals, etc. The ontology of ring subsets is not that different from an ontology of set systems.

Lattices are all well and good, in any non-trivial application you are not going to be considering any particular lattice but rather a whole hierarchy of closure systems (which form lattices). The same applies to binary relations, for example, there you have a whole hierarchy of closure systems like reflexive, symmetric, transitive, dependency relations, preorders, equivalence relations, and countless others that are not as commonly known.

Perhaps more interesting is the simplicity of polynomials over commutative rings as data structures. Here again there are similarities, for example a homogeneous polynomial is analogous to a uniform hypergraph, except instead of degrees you have cardinalities. The most important thing is that polynomials are such simple data structures. This makes polynomials as amenable to computations as any structure we are familiar with from combinatorics. This is why computational algebraic geometry is such an exciting field, with infinite potential.

On the algebraic level the most important common feature is commutativity. Commutative semigroups are a common framework for set theory, topology, lattice theory, commutative algebra, algebraic geometry, etc. I still haven't worked out the role of commutative semigroups in lattice theory, particularly non-idempotent posetal commutative semigroups, but I still believe that line of thought has unexplored potential. Commutativity clearly plays a fundamental role in the foundations of abstract algebra. If you will spare me yet another analogy this time pertaining to graph theory:

• Comparability graphs
• Commuting graphs

When orders are not total orders, we don't call them "nontotal" we call them partial. In the same vein, if a binary operation is not commutative they are not really "noncommutative" but rather partially commutative. There is not "noncommutative algebra" there is partially commutative algebra. At least that is the way I look at it. In the same way that comparability graphs determine the extent to which an order is "partial" commuting graphs determine the extent to which a binary operation is commutative. The first thing I think about when considering a semigroup is its commuting graph.

Todo: I need to post a sequel to my post on category theory, this time dealing with categorical logic. It will deal with subobjects, quotients, topoi, and the distributive lattices and heyting algebras associated with them. Other topics I need to post about are sheaves, localisation and schemes, posetal hypersemigroups, congruences of lattices, the ontology of lattices, a new congruence-based theory of permutation groups, additional aspects of the structure theory of commutative semigroups like the role of archimedean semigroups, and numerical semigroups, to name a few.