Maximal commuting cliques

One of the basic problems in semigroup theory is the relationship between subalgebras and commutativity. We will address this question by first considering commuting cliques of a semigroup. A commuting clique is a set of elements, each of which commutes with each other. In other words, a clique in the commuting graph of the semigroup. We will show that semigroup closure maps commuting cliques to commuting cliques.

Theorem. Let $S$ be a semigroup and let $C$ be a commuting clique of the semigroup. Then let $a,b$ be elements of $C$. The product $ab$ commutes with every element of $C$.

Proof. Let $c$ be any element of $C$. We will show that $(ab)c$ = $c(ab)$. By associativity we know that $(ab)c$ equals $a(bc)$. Then by the fact that $b,c$ are both elements of $C$ we know that they commute so $a(bc) = a(cb)$. By associativity we know that $a(cb)$ equals $(ac)b$. By the fact that $ac$ are both elements of $C$ we know this equals $(ca)b$ now finally by associativity this equals $c(ab)$. Applied iteratively this implies that semigroup closure maps commuting cliques to commuting cliques.

Corollary. the maximal commuting cliques of a semigroup form a subalgebra system

This follows from the fact that semigroup closure maps commuting cliques to commuting cliques. Suppose that $M$ is a maximal commuting clique, then by the fact that closure operations are extensive, the closure of $M$ must be greater then or equal to $M$ and by the fact that commuting cliques are mapped to commuting cliques, it must be mapped to a commuting clique greater then or equal to $M$. But $M$ is a maximal commuting clique by definition, so it must be mapped to itself. Therefore its closure must be equal to itself, so it must be a subsemigroup.

This theorem immediately gives us a whole class of subalgebras that can be infered immediately from the commuting graph of the semigroup, without having to consider the semigroup itself. The maximal commuting cliques form a subalgebra system, that is to say a subset of Sub(A), they also form a set system. This set system forms a maximal cliques family, and the underlying graph is the commuting graph. Therefore, in the other direction, the subalgebra system consisting of all maximal commuting cliques fully determines the commuting graph.

Corollary. The maximal commuting cliques of a semigroup form a maximal cliques family and the underlying graph of this family is the commuting graph.

This follows directly from the definition of maximal clique families. We can now add the maximal commuting cliques to our list of subalgebra systems that form interesting set systems. The maximal subgroups for example are a pairwise disjoint set system. In general, most different types of set systems emerge at one point or another from subalgebras.