Classes of divisibility commutative semigroups

Divisibility commutative semigroups are precisely the semigroups that have their L and R relations coincide. That is, they are the semigroups which act like commutative-semigroups with respect to divisibility, and they include all semigroups which are factorisation partially ordered. The consideration of these semigroups recently, as well as their specializations led to us to consider a number of subclasses of the class of divisibility commutative semigroups. An ontology of them is displayed below.

Clifford semigroups are particularly interesting because they are the most natural generalization of groups within the class of semigroups. In the theory of symmetric inverse semigroups, the Clifford semigroups are constructed entirely from charts that consist of only a permutation part and no nilpotent part. Clifford semigroups include both groups and semilattices as described above. We will now transition from our consideration of divisibility commutative semigroups to other generalisations of commutativity.

Strong divisibility commutative elements

We previously mentioned that semigroups can be strongly divisibility commutative. This occurs when the left and right principal ideals of the semigroup coincide for each element. In the case that the entire semigroup is not strongly divisibility commutative, there is a subset of the elements that are. So we will need to briefly examine strongly divisibility commutative elements and what they mean for Green's relations.

Definition. an element of a semigroup is strongly divisibility commutative if its left and right principal ideals coincide. The strong divisibility center consits of all elements that are strogly divisibility commutative.

It is trivial to see that central elements are strongly divisibility commutative, and therefore the center is a subset of the strongly divisibility center. Now consider the effect of strong divisibility upon the Green's relations. If two elements are strongly divisibility commutative, then they divisibility commute in the sense that L and R for the two elements are logically equivalent.

Theorem. if a and b are both strongly divisibility commutative then a L b is logically equivalent to a R b.

Proof. this will be proved by demonstrating implication in both directions. Suppose that a L b then we know that L(a) = L(b) but by strong divisibility commutative of elements we have R(a) = L(a) and L(b) = R(b) which implies R(a) = L(a) = L(b) = R(b) which by transitivity means that R(a) = R(b) which by definition means a R b. In the opposite direction a R b implies that R(a) = R(b) which implies that L(a) = R(a) = R(b) = L(b) which similarily implies L(a) = L(b) which by definition means a L b. So the proof of implication in both directions implies logical equivalence.

I managed to use this and Green's theorem to prove that idempotent-central semigropus (which include Clifford semigroups) are L,R idempotent separating. The H classes are also idempotent separating by Green's theorem, but there is no obvious way to get that the D classes are idempotent separating so we can't prove that.

Theorem. idempotent-central semigroups are L,R idempotent separating

Proof. let a and b be two idempotents of the semigroup, now a and b are both in the center by definition so they are strongly divisibility commutative. This implies that if a L b then a R b and that if a R b then a L b, so if idempotents are L related or R related then they are both L and R related, which would mean that they are H related. But by Green's theorem two idempotents cannot be H related, which would be a contradiction. So L and R must be idempotent separating.

Strongly divisibility commutative semigroups

We previously addressed the idea of divisibility commutative semigroups. These are semigroups in which the Green's L relation and the Green's R relation coincide. But it is apparent, even from the simplest case of the non-commutative divisibility commutative semigroup on three elements, that having the Green's L relation and the Green's R relation coincide does not imply that left and right principal ideals are going to be the same.

Definition. a semigroup is called strongly divisibility commutative if left and right ideals coincide.

The first thing is to prove that these strongly divisibility commutative semigroups are in fact divisibility commutative, for the sake of formality.

Theorem. strongly divisibility commutative semigroups are divisibility commutative

Proof. suppose that the L class of x and the L class of y are equal, then Sx = Sy, so since Sx = xS and Sy = yS we have that xS = Sx = Sy = yS which implies that xS = yS by the transitivity of multiplication. This means that x,y are in the same R class. In the reverse direction, if x,y are in the same R class then Sx = xS = yS = Sy implies that x,y are in the same L class. So R = L and the two Green's relations coincide when the principal ideals coincide.

It is trivial that commutative semigroups are strongly divisibility commutative. It is also the case that groups are strongly divisibility commutative. The interesting thing, is that there is a specific method to get the element that produces a given output element in the other principal ideal: namely conjugation of group elements.

Theorem. in a group if gx = yg is an element in both the left and right principal ideals of a given element g then x,y are conjugates of one another

Proof. if we take gx = yg and we multiply each side by g^(-1) to the front then we get x = g^(-1)yg which means that x is a conjugate of y. In the other direction, if we multiply each side by g^(-1) to the back we get gxg^(-1) = y and that means that x and y conjugates.

So conjugation in group theory actually is the basis of the strong divisibility commutativity. It is not hard to see then, why it is the case that self-conjugate elements are centered in groups, because conjugates are the elements that produce a given result in the opposite direction, and when this result is always the element must commute. It is known that in group theory, conjugates are directly related to commutativity in that for example the size of a conjugacy class is the order of the group divided by the commuting degree.

Previously we talked about Clifford semigroups, which are known to be the only divisibility commutative regular semigroups. According to the encyclopedia of mathematics, Clifford semigroups are strongly divisibility commutative as well because L and R classes coincide. I am willing to conjecture that since Clifford semigroups are completely regular, the element that produces a given output with respect to an element in the opposite argument order is the internal conjugate of that element in its H class. But I don't have a proof or a counter-example yet. The simpler case of groups is illuminating nonetheless.

Group-symmetric semigroups

In the previous post the nature of Clifford semigroups was briefly discussed. We noticed that Clifford semigroups are divisibility commutative (because L=R). But due to complete regularity the Green's relations of Clifford semigroups have the further property that all non-trivial H classes form subgroups. We can therefore form a special subclass of the class of divisibility commutative semigroups that captures the divisibility properties of Clifford semigroups.

Definition. a semigroup is called group-symmetric if it is divisibility commutative and there are no non-trivial H classes that do not form groups.

The reason that I call these semigroups "group-symmetric" is that all the non-trivial components of the factorization preordering are a result of subgroups. Since subgroups are responsible for all factorization symmetry, antisymmetry is equivalent to aperiodicity. Here are three basic theorems about these semigroups:

• Clifford semigroups are group-symmetric
• All commutative semigroups of order four or less are group-symmetric
• Finite monogenic semigroups are group-symmetric

To see (1) first recall that Clifford semigroups are divisibility commutative by a previous theorem. Then by complete relugarity all H classes form subgroups, so there are no non-trivial H classes that do not form groups. To see (2) consider that Green's theorem demonstrates that a non-trivial H class that does not contain an idempotent must not contain the iterations of any of its own elements, so there must be a third element that is an element of these two. Additionally, the semigroup cannot be aperiodic because then it would be H-trivial since it is finite. So there needs to be a group which means at least two other elements need to exist to contain the group leading to a minimum of five elements. To see (3) compute the H classes of the finite monogenic semigroup. Clearly there is an H class containing the idempotent, but then every other element is H-trivial because a lesser iterate cannot be obtained from a larger one outside the group.

Starting with order five, there are commutative semigroups which are not group-symmetric. But these semigroups are not group-free because it is proven that all finite commutative aperiodic semigroups are antisymmetric and hence trivially group-symmetric. So even in those cases the symmetry in the semigroup is because of some subgroup. Those non-trivial H classes that are not subgroups are externally symmetric because they emerge from some group outside of themselves which operates on them to create some symmetry in the semigroup, which absent everything else would be antisymmetric like a finite commutative aperiodic semigroup.

Definition. the group elements of a semigroup are all those that are contained in some subgroup and the non-group elements are all those elements are not contained in a subgroup

The order of any finite semigroup is equal to the sum of the group elements count and the non-group elements count. So for example, example monogenic semigroups are classified by their period and index. The only difference is that their sum doesn't equal to the order of the semigroup because the index is never zero even for cyclic groups. So the non-group elements count is equal to the index minus one. The non-group elements count determines how far the semigroup is from being completely regular, and therefore in the group-symmetric case from being Clifford.

We saw how, particularly in the finite case, J-trivial semigroups generalize semilattices. Group-symmetric semigroups allow us to do the same thing for Clifford semigroups, which are often called semilattices of groups. So for example, given a group-symmetric semigroup we can form a Clifford semigroup from it which contains its ordered group structure. In the opposite direction, given a semilattice of groups we can form different group-symmetric semigroups which maintain the Cliffordic structure of that semigroup.