The lattice of semigroup varieties

Semigroups are considered here in terms of the identities which they may satisfy with the emphasis on the identities and on classes of semigroups defined by identities, rather than on the semigroups themselves. These classes, called varieties, form a lattice under inclusion and it is illuminating to interpret properties of semigroup identities and varieties in terms of this lattice, its cardinality, its atoms, its infinite chains, and so on. This paper is an expansion of addresses to the Southeastern Section of the Mathematical Association of America at East Carolina University, March 1968 and to the Symposium on Semigroups and Rings at the University of Puerto Rico, Mayaguez, March 1970. The preparation of the paper was supported in part by NSF Grants GP 6597 and GP 20638.