Rees factors of J-trivial semigroups

The arithmetical properties of finite boolean algebras and partition lattices are Rees factor semigroups of addition and multiplication. This naturally lends us to a further study of the Rees factor semigroups of J-trivial semigroups. J-trivial semigroups are subalgebra closed, but not quotient closed so a first step is to check if Rees factor semigroups always preserve J-triviality.

A necessary condition in that direction is to determine if all the congruence classes of a Rees semigroup congruence in a J-trivial semigroup are convex. It is common when dealing with ordered algebraic structures to require congruence classes are convex, for example all congruence classes of a lattice congruence are convex. This is an easy verification.

Proposition. let $S$ be a J-trivial semigroup and $I$ an ideal of $S$ then the congruence classes of the Rees semigroup congruence $C$ associated to $I$ are all convex.

Proof. (1) singleton sets are convex (2) ideals are convex because if $a,b \in I$ such that $a \subseteq c \subseteq b$ then $a \subseteq c$ implies $c$ is in $I$. So congruence classes are convex. $\square$

It is an easy verification that congruence classes of the Rees semigroup congruence are convex. It is a bit harder to show that their quotient is J-trivial.

Theorem. let $S$ be a J-trivial semigroup and $I$ an ideal then $\frac{S}{I}$ is J-trivial with partial order $I^C + 0$.

Proof. (1) suppose that $a,b \in I^C$ then $a \subseteq b$ in $\frac{S}{I}$ provided that $\exists x,y \in (\frac{S}{I})^1 : xay = b$. Then suppose either $x$ or $y$ is $I$ then $xay = b$ implies that $I \subseteq b$ which means that because $b$ is a filter than $b \in I$ but that is a contradiction because we had that $b \in I^C$ so $x,y \in I^C$ which means that the order of $I^C$ is a suborder of $S$.

Then to prove that the ideal $I$ is an adjoined zero element in $\frac{S}{I}$ we need to prove two conditions (1) $I$ is not less then any element in $I^C$. This follows because $I$ is an ideal, so it includes all its successors. (2) every element is less then $I$ in $\frac{S}{I}$. This one follows from the fact that J-trivial semigroups are directed posets, so every element has a successor in an ideal. $\square$

As this proof involves the characterization of the algebraic preorders of Rees factor semigroups, we can translate this into order theoretic terms to get the Rees factors of a preorder directly. In general, its good to have our terminology from order theory and semigroup theory coincide.

Definition. the Rees factor of a preorder with respect to an upper set (respectively a lower set) is the quotient order with respect to ordinary precedence produce by collapsing the upper set (respectively a lower set) into a single element and preserving every other element.

The partial orders of finite J-trivial semigroups are not always semilattices. Therefore, it is not an entirely trivial result that the Rees factors of $(\mathbb{Z}_+,*)$ by total order upper sets are semilattice ordered.

Proposition. The Rees factor semigroups of $(\mathbb{Z}_+,*)$ by total order upper sets $\{x : x \ge n\}$ are semilattice ordered.

The partition lattices are important because they provide the logical foundations for the semantics of multiplication. We can now characterize some of the properties of the arithmetic properties of finite partition lattices by Rees factors.

Proposition. let $\frac{(\mathbb{Z}_+,*)}{\ge n}$ be the Rees factor semigroup of positive integer multiplication. Then it has at most two idempotents (1 and $n$) and every other element has index equal to the number of powers strictly less then $n$ plus one.

Proof. (1) $n$ and the $\ge n$ class are preserved as idempotents (2) let $x$ be any other element then $x^m$ is equal to $n$ provided that $x^m \ge n$ so the index of $x$ is the number of $x^m \not= n$ plus one for $x^m = n$. $\square$

Additional properties of these Rees factor semigroups could be determined analytically, such as by applying the prime number theorem, so a variety of methods can be applied to study Rees factors of the positive integers multiplication semigroup.