The Class of Structurally Regular Semigroups

A semigroup S is said to be structurally regular if there exists an ordered pair (n; m) of non-negative integers such that the quotient S=(n; m) is regular in the usual sense. The congruence (n; m) may be dened in the following manner: (n; m) = f(a; b) : uav = ubv; 8u 2 Sn; v 2 Smg; where Sk for k 1 denotes the set of all products of k elements in S, while S0 denotes the singleton set containing only the identity element of the monoid S(1) obtained by adjoining an identity element to S. Alternatively, a semi-group S is structurally regular if there exists some ordered pair (n; m) such that for each element a 2 S there exists an element b 2 S (not necessar-ily unique) such that for all u 2 Sn and for all v 2 Sm the two equalities uabav = vav and ubabv = ubv hold. The concept of structural regularity was rst introduced in [2], and is a generalization of the usual concept of regularity, but is distinct from othe