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