Divisibility commutativity in the sense of green's relations

Given a commutative semigroup, then that semigroup has an associated preorder $a \le b$ if there exists a c such that $ac = b$. This is a preorder that defines all the ordering properties of a commutative semigroup in a singular standard way. The situation is not so simple in non-commutative semigroups as seen by the different green's relations. But suppose that L = R then we know that since D and H are defined by the meet and join of L and R in the partition lattice this means that all four are equal. \[ L = R = D = H \] In these cases, we can instead focus on a single preorder. Well there are different preorders on a non-commutative semigroup (the Green's preorders each have L,R,J,and H as partitions) these preorders are also equal in a divisibility commutative semigroup. That is, there is a sense that the semigroup has a singular order just like commutative semigroups. We already discussed J-trivial semigroups which are essentially just semigroups on total orders, well semigroups with a symmetric total preorder are essentially groups (which is proven in the finite case).

• Antisymmetric: J-trivial semigroups
• Symmetric: Groups

It makes sense that groups are divisibility commutative, because everything can divide everything else in both directions. So we see that the two basic constituents of divisibility commutative semigroups are J-trivial semigroups and groups. J-trivial semigroups are aperiodic well groups are periodic respectively. This includes non-commutative groups as well, as their non-commutativity doesn't effect their divisibility. In general, divisibility commutative semigroups allow a restricted amount of non-commutativity so long as it doesn't effect divisibility.

The theory of J-trivial semigroups

Starting from partial orders, we can form semilattices from partial orders satisfying certain conditions. Semilattices are restricted to semigroups that are commutative and idempotent. Relaxing the restriction of idempotence, we arrived at finite commutative aperiodic semigroups. The next step, of relaxing commutativity, produces J-trivial semigroups. It can be proven that finite commutative aperiodic semigroups are J-trivial. We already described the simplest J-trivial semigroup that is not commutative, the T3 special combiner on a totally ordered set of three elements.

These non-commutative J-trivial semigroups produce upper bounds of elements in a partial order in an argument-dependent manner. This opens up a whole wide range of possibilities that were not available previously and it gives an approach to considering different upper bound producing functions on partial orders. Actually, J-trivial semigroups constitute all associative upper bound producing functions on partial orders. As we discussed previously, there are two types of non-monotonic behavior that can occur on J-trivial semigroups: chain non-monotonic behavior between comparable elements and antichain non-monotonic behavior on incomparable elements.

• Chain non-monotonic: non-monotonic products of comparable elements
• Antichain non-monotonic: non-monotonic products of comparable elements

We already demonstrated examples of chain non-monotonic and antichain non-monotonic behavior. These can occur argument-order dependently, essentially what this means is that given two elements which are either related to one another or not related to one another by the partial ordering comparability relation, these elements can produce a larger result in one argument order then in another argument order. This makes it so that one argument order is greater then another argument order.

In larger semigroups, smaller semigroups like the totally ordered T3 combiner can be combined in different directions, so they are not isomorphic even though they are antiisomorphic. In such semigroups for certain elements one argument order can be greater then another, well in other elements a greater result is produced between them by arguments in a different order. In other cases they can be combined in the same argument order. So these non-commutative J-trivial semigroups can either favor one argument order or another, it it can maximize them both equally. This is one thing that adds to the complexity of J-trivial semigroups.

Non-commutative version of the exceptional commutative semigroup

The exceptional small commutative semigroup of order four was defined primarily by the different product of its incomparable minimal elements. Rather then then producing the least upper bound it produced the maximal element of the partial order. Amongst the non-commutative semigroups there is a similar semigroup that produces either the maximum or the join depending upon the argument order.

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 2, 1 ], 
  [ 1, 1, 2, 2 ] ]

So in total for the same indices and the same factorisation partial order there are three different types of behavior: the semilattice like behavior, the maximizing behavior, and the argument dependent half way case. All of these semigroups, just like the T3 special semigroup on three elements discussed previously are J-trivial. So we will be considering J-trivial semigroups which are associative bound-producing functions on partial orders.

Smallest non-commutative totally ordered semigroup

We considered commutative aperiodic semigroups such as semilattices upper bound functions on partial orders. We even considered how these upper bound functions can themselves be partially ordered by "how upper" their bounds are. Now we will move to the non-commutative case. Towards that end, we should consider the simplest case of a non-commutative upper bound function:

[[1 1 1]
 [1 1 1]
 [1 2 3]]

The first thing we notice is that this semigroup is actually totally ordered [3,2,1]. The second thing that we notice about this semigroup is that the middle element is index two. Therefore, the corresponding monotonic semigroup is M2,1 + identity or the index two aperiodic monogenic semigroup with an identity element adjoined to it.

We can learn about this semigroup by comparing it to its monotonic commutative aperiodic counterpart. The only difference is that 2*3=1 rather then 2 this means that the middle and minimal elements can either produce their least upper bound which is the middle element or the maximal element. Since this produces a greater upper bound then it would otherwise would among comparable elements I call this chain non-monotonic. This special T3 semigroup often appears in larger partially ordered semigroups as a subsemigroup.