\(M\)-classification of regular semigroups (Q1865679)

The author proposes a scheme to classify regular semigroups and takes a first step in that direction. On the congruence lattice of any regular semigroup \(S\) are defined two relations, \(K\) and \(T\), the latter a congruence, the former only meet-preserving, in general. Congruences are \(K\)-related if they share the same kernel -- union of idempotent classes -- and are \(T\)-related if they share the same trace -- restriction to the idempotents of \(S\). The greatest congruences sharing the same trace and kernel, respectively, with the identity relation on \(S\) are denoted \(\mu\) and \(\tau\); the least bearing the same relationships with the universal congruence on \(S\) are denoted \(\sigma\) and \(\beta\). These four congruences have been intensively studied. They generate a sublattice \(\Gamma\) of the congruence lattice of \(S\). The proposed classification is via a category modeled on the triples \((\Gamma,K,T)\), together with rather naturally defined morphisms. In this paper a simpler situation is treated, whereby the lattice generated only by \(\sigma\), \(\tau\) and \(\beta\) is considered in a similar way. It is shown that there are 16 isomorphism classes of objects in the corresponding abstractly defined category of triples and that each can be realized as the concrete triple associated with a certain regular semigroup.