### The Lattice of Equational Classes of Commutative Semigroups

Canadian Journal of Mathematics, Vol. 23, No. 5 (1971), 875-895

Received May 19, 1970. This paper is a revision of the author's doctoral dissertation, written under the supervision of Dr. G. Bruns at McMaster University, Hamilton, Ontario.

Introduction

There has been some interest lately in equational classes of commutative semigroups. The atoms of the lattice of equational classes of commutative semigroups have been known for some time. Perkins has shown that each equational class of commutative semigroups is finitely based. Recently, Schwabauer proved that the lattice is not modular, and described a distributive sublattice of the lattice.

The present paper describes a "skeleton" sublattice of the lattice, which is isomorphic to A × N^{+} with a unit adjoined, where A is the lattice of pairs (r,s) of non-negative integers with r ≤ s and s ≥ 1, ordered component-wise, and N^{+} is the natural numbers with division. Every other equational class "hangs between" two members of the skeleton in a certain way; the relationships between intervals of the form [R_{1}, R_{2}] where R_{1}, R_{2} are members of the skeleton are investigated. Finally, it is shown that Schwabauer's distributive sublattice is actually a maximal modular sublattice.