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.