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

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