Commutativity-necessary subsemigroups overview

Let $G$ be the commuting graph of a semigroup. Then the following subsets of the commuting graph $G$ are necessarily subsemigroups of any semigroup that has it as its commuting graph.

• Adjacency sets
• Adjacency princpal filters
• Maximal cliques
• The set of all universal vertices
• The intersections of any of these

These concepts are then translated into centralizers, commutative principal filters, maximal commuting cliques, and the center when expressed in algebraic terms rather then graph-theoretic ones. All of these are ensured to be subsets for any semigroup with a given commuting graph. If instead we take a magma, which also has a commuting graph, then none of these subsets are ensured to be closed. Centralizers don't always form submagmas, nor is it the case that monogenic magmas are necessarily commutative. These subsemigroups simply tell us how associativity effects commutativity.

Examples:

Net graph:

The elements 0,1,2 are commutativity-maximal and therefore they are idempotent, furthermore {0,1,2} forms a maximal clique so it is a subsemigroup. Therefore, {0,1,2} forms a subsemilattice of any semigroup with this commuting graph. In this way, we can see how subsemilattices can be inferred from the commuting graph of a semigroup. There are two possible semilattices on three elements that this could be. The sets {0,3},{1,4}, and {2,5} are also maximal cliques and are therefore subsemigroups. Finally, {0,1,2,3},{0,1,2,4}, and {0,1,2,5} are centralizers.

Co-net graph

The sets {0,1,2},{0,1,3},{0,2,4}, and {1,2,5} all form maximal cliques. If we intersect {0,1,2} with any of the other three we get {0,1},{1,2},{0,2} and then if we intersect those we get the commutative principal filters {0},{1},{2} which once again shows that {0,1,2} is a subsemilattice. Except now in this case, we know that the subsemilattice must be the semilattice on a total order of three elements. The three centralizers in this case are {0,1,2,3,4},{0,1,2,3,5}, and {0,1,2,4,5}.

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Small graphs