The lattice of r-unipotent congruences on a regular semigroup

Abstract

If S is an inverse semigroup then E is a congruence on C(S). If S is a regular semigroup then each E -class of C(5) is a complete modular sublattice of C(5). (See [6]. ) In [5, Sec. 3] Petrich presents a few characterisations of E when S is an inverse semigroup. Here we prove that on a regular semigroup 5, the relation E restricted to RC(5) is a congruence. Also we extend Petrichfs results to the lattice RC(S) of a regular semigroup 5 and present a characterisation of the greatest element of each E-class. Characterisations of the least element of each E-class have been presented by Feigenbaum [I] and La Torte[4]. THE LATTICE RC(S) We use, whenever possible, the notation of Howie [3]. Recall first that a regular semigroup 5 is said to be R-u_~potent if its set of idempotents E(S) is a left reqular band, i.e. if E(5) satisfies the identity ere = el. In [7,1; 8, 1.1 ] it is shown that on a regular semigroup 5,