Taras Banakh – A semigroup is finite if and only if it contains no infinite chains and no infinite antichains

Ivan Franko National University of Lviv
A subset $A$ of a semigroup is called a chain (resp. antichain) if for any (distinct) elements $x,y\in A$ the product $xy$ does (not) belong to $\{x,y\}$. We prove that a semigroup $S$ is finite if and only if $S$ contains no infinite chains and no infinite antichains.