Directed preorders and semigroups

It should be readily apparent that not all posets can appear as the orderings of J-trivial semigroups. By the same token, not all preorders can arise from semigroups in general. A first thought is that perhaps all orderings of J-trivial semigroups are semilattices, but of course this isn't the case for semigroups of order greater then four: because there is a unique [2,2,1] weak ordered commutative J-trivial semigroup. What we can get is a generalisation of semilattices: the class of directed posets.

Theorem. let $S$ be a semigroup then its algebraic $J$ preorder $\subseteq$ is a directed preorder, which means that every two elements $a$ and $b$ have an upper bound.

Proof. let $a,b \in S$ then we can construct an upper bound by $ab$ $\square$

Corollary. every J-trivial semigroup is a directed poset.

Directed posets can be seen to be the posets that have some kind of compositional process associated to them, even if that compositional process is not a semilattice. Upper bounds correspond to the composition of elements.

Example. the free non-commutative semigroup $S$ on a set of generators $X$ is a J-trivial semigroup, whose partial ordering is the consecutive subsequence ordering. This is not a semilattice, but it is directed because any two words have an upper bound provided by their composition.

The example of the free non-commutative semigroup should demonstrate that $L$ and $R$ preorders are not directed in general, even though $J$ is. The prefix and postfix orderings on the free non-commutative semigroups are trees, so elements don't have an upper bound unless they are comparable. There is also the example of a pure rectangular band, which has one of its $L$ or $R$ preorders an antichain.

The reason every pair of elements in a semigroup has an upper bound, is every element has a composition that exists. It follows that in a partial semigroup such as a category, its preorder need not be directed. Morphisms in different connected components don't need to have an upper bound.

• The $L$ and $R$ preorders of a semigroup need not be directed.
• The morphism preorder of a category need not be directed.

A directed preorder describes a set of elements for which a compositional process exists, but those don't need to exist in a partial semigroup such as a category. Nonetheless, we can use directed posets in the study of J-trivial semigroups. Directed posets still have suprema they just don't need to be unique, so we have to adjust the theory a little bit.