A Semigroup Is Finite Iff It Is Chain-Finite and Antichain-Finite

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