Variants of the lattice of partitions of a countable set

In this paper we consider the ordered by inclusion lattice $\operatorname{Part}(M)$ of all partitions of a countable set $M$. The lattice $\operatorname{Part}(M)$ 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 $\operatorname{Part}(M)$.