Variants of a lattice of partitions of a countable set

Abstract

In this paper we consider the ordered by inclusion lattice $\text{Part}(M)$ of all partitions of a countable set $M$. The lattice $\text{Part}(M)$ is a semigroup with respect to the operation $\wedge$ which maps two partitions to their greatest lower bound. We obtain necessary and sufficiency conditions for isomorphism of two variants of the semigroup $\text{Part}(M)$.