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

Iryna Banakh, Taras Banakh, Serhii Bardyla

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 $x,y\in S$. A semigroup $S$ is called ($anti$)$c…