A semigroup is finite if and only if it is chain-finite and antichain-finite

TL;DR

This paper characterizes finite semigroups as those that are both chain-finite and antichain-finite, establishing new properties of antichain-finite semigroups and providing a counterexample involving semilattices.

Contribution

It proves that antichain-finite semigroups are periodic with finite roots of idempotents and characterizes finiteness via chain- and antichain-finiteness, including a counterexample.

Findings

Antichain-finite semigroups are periodic.

For each idempotent, the set of roots is finite.

A semigroup is finite iff it is both chain- and antichain-finite.

Abstract

A subset $A$ of a semigroup $S$ is called a $chain$ ($antichain$) if $xy\in\{x,y\}$ ($xy\notin\{x,y\}$) for any (distinct) elements $x,y\in S$. A semigroup $S$ is called ($anti$)$chain$-$finite$ if $S$ contains no infinite (anti)chains. We prove that each antichain-finite semigroup $S$ is periodic and for every idempotent $e$ of $S$ the set $\sqrt[\infty]{e}=\{x\in S:\exists n\in\mathbb N\;\;(x^n=e)\}$ is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Also we present an example of an antichain-finite semilattice that is not a union of finitely many chains.