Generalizing semilattices

The generalizations of semilattices should be commutative and aperiodic (or group-free) because these semigroups are most similar to semilattices. Together, these two conditions ensure that every finite semigroup will have an algebraic preorder that is antisymmetric, that is that the elements of the semigroup are partially ordered. This isn't true for non-aperiodic semigroups, for example, abelian groups are obviously symmetrically ordered as every element can be acquired from every other one due to reversibility. It also isn't true for even the simplest non-commutative semigroup which is also a band as well as aperiodic.

A brief examination of commutative aperiodic semigroups reveals that not all of them tend tend to have the most semilattice-like properties. The first aperiodic commutative semigroup that I noticed did not look like a semilattice is the GAP small semigroup 4, 34 which doesn't preserve order joins over distinct elements of its minimal generating set. It is weak ordered by [2 1 1] and the distinct minimal elements go to the maximal element, rather then the middle which is also the least upper bound with respect to that algebraic ordering. This leads to the following definition.

Definition. a commutative aperiodic semigroup is order-preserving if it has a presentation over a set of generators such that the product of two elements a*b is equal to the join (least upper bound) of the algebraic preorder of the semigroup.

This leads to a broad generalization of semilattices within the category of commutative aperiodic semigroups. Then to consider the semilatticeness of a commutative aperiodic semigroup one could consider the number of relations that need to be added to the presentation in order to define the semilattice besides the order ones which are simply assumed.

This leads to the notion of commutative aperiodic semigroups which can be determined by multiset systems, rather then by set systems as semilattices are. Semilattices are defined by their join representations, which produce join representation families in the theory of set systems, for example, finite power sets correspond to the join representations of boolean algebras.