Commutative principal filters

We will now use the commutativity preordering to produce certain subsemigroups of a semigroup. The principal filter of an element in a preorder is the set of all elements greater than or equal to it. The commutative principal filters we will be examining are a special case.

Definition. Let $x$ be an element of a semigroup. Then the commutative principal filter of $x$ consists of all elements of $x$ that are greater than or equal to it in the commutativity preordering $\{ y : \forall a: (xa=ax) \implies (ya=ay) \}$.

We will prove that commutative principal filters are subsemigroups by demonstrating that they are precisely the intersection of all maximal cliques containing an element. Which makes it a subsemigroup by our theorem on the intersection of maximal cliques.

Lemma. Suppose that S is a semigroup, x is an element, and C is the intersection of all maximal cliques containing x. Then every element y of C is commutatively greater than or equal to x.

Let $a$ be an element of S that commutes with x. Then a,x forms a clique but not necessarily a maximal clique. Every clique is a subset of some maximal clique, so there is some maximal clique M that contains a and x, but since y is in the intersection of all maximal cliques it is in every maximal clique including M. Therefore, since y is in M it commutes with a. Therefore, every element a that commutes with x must commute with y, which means that y is commutatively greater than or equal to x.

Lemma. let b be commutatively greater than a, then all the maximal cliques of a contain b

Suppose we have a maximal clique M containing a. By the definition of supercommutativity, we know that b must commute with everything that a does. That means that there is some clique that contains M as well as b. Supposing that M contains b, then we have what we aimed for, if on the other hand it doesn't contain b, we have a greater clique that does which contradicts the condition of maximality. So by contradiction, b is in all the maximal cliques of a.

This demonstrates that the commutative principal filters are equal to the intersection of all maximal cliques containing an element, and that the intersection of all maximal cliques containing an element are the commutative filters. As they correspond in both directions, they are equal.

Lemma. the commutative principal filter and the intersection of all maximal cliques containing an element are equal.

We proved separately that maximal commuting cliques are subsemigroups, and that the intersection of subsemigroups is a subsemigroup, so now we can get to the main point. By the fact that commutative principal filters are the intersection of maximal commuting cliques they are subsemigroup.

Theorem. commutative principal filters are subsemigroups.

This has an immediate corollary, which is useful in studies of commuting graphs. Given a commuting graph of some semigroup, we know that the commutativity-maximal elements are idempotent. This can also be infered by the fact that the commutativity preordering is a superpreordering of the generator preorder, but in either direction it is true.

Theorem. commutativity maximal elements of semigroups are idempotent

Maximal elements in a preorder have no other elements greater than or equal to them, therefore they have a singleton principal filter which leads to a singleton subsemigroup. A singleton subsemigroup contains only a single idempotent element. An immediate result of this is that every semigroup with a commuting graph equal to the cyclic graph C4 is a band. Actually, there only two such semigroups produced by ordered combination of anticommutative bands on two elements that are either isomorphic or anti-isomorphic to one another. The same principle applies to the cyclic graph C5, the complete bipartite graph K2,3, and any other connected triangle-free graph that contains no leaf vertices.