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 \(\text{Part}(M)\).