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

A subset A of a semigroup S is called a chain (antichain) if ab∈{a,b} (ab∉{a,b}) for any (distinct) elements a,b∈A. A semigroup S is called periodic if for every element x∈S there exists n∈N such that xn is an idempotent. 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 e∞={x∈S:∃n∈N(xn=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. Furthermore, we present an example of an antichain-finite semilattice that is not a union of finitely many chains.