Finite commutative totally preordered semigroups

The subject of this examination is the location of the non-trivial non-group H classes in a commutative semigroup. This can be used to solve the inverse condensation problem of which semigroups have a given finite total order commutative condensation type $\frac{S}{J}$. This naturally follows from semilattice decompositions, which reduce such semigroups into their Archimedean components.

A consequence of this is that we can completely characterize all commutative totally preorder semigroups by semilattice decompositions. This naturally generalizes to condensible totally preordered semigroups with commutative condensation, by replacing the J-classes with other simple semigroups. Generalizations of the structure theorem will be briefly considered.

Lemma 1. non-trivial group $H$ classes in commutative semigroups are enclosed

Proof. (1) suppose that a given non-group $H$ class is maximal then $H^2 \cap H = \emptyset$ means that $H$ iterates to produce something greater then it, but it is maximal so there is nothing for it to produce which is a contradiction. (2) suppose that a given non-group $H$ class is minimal then since $H^2 \cap H = \emptyset$ the action representatives moving elements around in $H$ must come from its predecessors, but no such predecessors because it is minimal. $\square$

This is why commutative semigroups of order four or less are all group symmetric. The existence of non-trivial non-group $H$ classes necessitates the existence of extra elements to inclose the $H$ classes.

Lemma 2. let $S$ be a commutative semigroup with finite monogenic condensation $\frac{S}{H}$, then all non-maximal $H$ classes in $S$ are singletons.

Proof. let $C$ be a non-maximal $H$ class in $S$, then since $S$ has monogenic condensation all non-maximal $H$ classes are non-groups. Suppose that $C$ is non-trivial and has at least two elements, then there is some action representative predecessor of it that moves any given element $x \in C$ to any other element $y \in C$. This means that there is some other $H$ class $D$ such that $CD = C$.

Naturally, absorpotion $CD = C$ means that $CD^n = C$. Every finite monogenic semigroup generates an idempotent, so the condition that $CD^n = C$ means that there must be some idempotent predecessor of $C$, but there are no idempotent predecessors of an element in a finite monogenic J-trivial semigroup. So $C$ must be a singleton. $\square$

We can now apply this lemma to totally preordered commutative semigroups to produce the structure theorem. This makes use of semilattice decompositions, condensation, and the characterization theorem for finite totally ordered commutative semigroups.

Theorem 1. a finite commutative totally preordered semigroup is a semilattice of Archimedean commutative semigroups with total preorder types [1 1 1 ... n]

Proof. $S$ is a commutative totally preordered semigroup so can be condensed $\frac{S}{H}$ into a finite commutative totally ordered semigroup. This is then a semilattice of totally ordered monogenic semigroups. Each of the Archimedean components in this semilattice is a semigroup with monogenic condensation as in lemma 2, and so all non-maximal $H$ classes are singletons. $\square$

A finite Archimedean commutative totally preordered semigroup can be identified with the data of a finite commutative group $G$ and an element of a given index that generates some cyclic subgroup of $G$. Finite monogenic semigroups are precisely the commutative totally preordered semigroups with an element that generates the entire group $G$.

Lemma. let $S$ be a monogenic semigroup, then $S$ is a totally preordered semigroup

Proof. let $x$ be the generator for $S$ then every element is of the form $x^n$ for some $x \in \mathbb{N}$. Then we have a total order $x \subseteq x^2 ... $ containing all the elements of $S$, so the algebraic preorder of $S$ must be a total preorder. $\square$

We can now construct an ontology: there are a number of classes of commutative semigroups dealt with in this classification determined by the layers of the decomposition provided by condensation, semilattice decomposition, and the fundamental theorem of finite commutative groups. The class of Clifford finite commutative totally preordered semigroups, is the class of linear semilattices of groups. In that case, the only relevant detail not covered by the structure theorem is the action of predecessor groups on their successors, which can actually be reduced to the action of a predecessor group to its parent's identity element.

This can be generalized first to divisibility commutative semigroups with commutative total order condensation, by replacing the finite commutative groups with any finite groups in general. Then secondly, it could be generalized to all totally preordered semigroups by replacing groups with other J-total semigroups like rectangular bands.