Irreducible varieties of commutative semigroups

In this paper we describe the varieties of commutative semigroups that are meet- and join-irreducible in the lattice of the varieties of commutative semigroups. We apply the method of A. Kisielewicz [Trans. Amer. Math. Soc. 342 (1994) 275–305]. This leads to investigation of the covering relation in the lattices of remainders and the algebraic structure of the remainders, involving permutation groups acting on the sequences of positive integers. In particular, along the way, we prove a theorem about existence of unique minimal generators for remainders, and provide algorithms to determine all the covers and dual covers of a given variety of commutative semigroups.