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

5 pages

References in corpus (1)

• Characterizing chain-compact and chain-finite topological semilattices