Cancellative Semigroups

Abstract

The semigroups indicated in the title (i.e. satisfying the two-sided cancellation law xa = xb → a = b, ax = bx → a = b) form a popular class of semigroups; it has attracted attention from the beginning of semigroup-theoretic investigations. Naturally, these semigroups were in the field of attention among the first classes of semigroups in investigations of lattice properties (in particular, of lattice isomorphisms) as well. Of course, groups are especially distinguished among cancellative semigroups; Section 34 is devoted to lattice isomorphisms of groups. Since in a periodic group there are no (non-empty) subsemigroups which are not groups, the semigroup specificity is of essence in relevant considerations only in the case of aperiodic groups. For a long time the following general question was open: is any aperiodic group [strictly] lattice-determined? Quite recently this question was negatively settled, see Subsection X.10; the results of Section 34 exhibit a number of large classes of lattice-determined groups.