Modularity of the lattice of congruences of a regular ω-semigroup

Abstract

Dedicated to Professor C. Tibiletti Marchionna on her 70th birthday In this paper a characterization of the regular co-semigroups whose congruence lattice is modular is given. The characterization obtained for such semigroups generalizes the one given by Munn for bisimple co-semigroups and completes a result of Baird dealing with the modularity of the sublattice of the congruence lattice of a simple regular co-semigroup consisting of congruences which are either idempotent separating or group congruences.

Semimodularity of the congruence lattice on regular ?-semigroups

“…Remark 3.4. It easy to show that the condition given in [3] on a simple regular ~o-semigroup for having a modular lattice of congruences implies the previous ones. It is also straightforward to prove that if d = 1 (i.e.…”

“…Thus we have tr (~/~ 2) = ~ A v with ~ A ~ < # and ~/x ~ < ~,, hence, by (ii) of Lemma 2.6, # ^ ~ -<; ~, r ^ ~ M v and A ~ ^ ~ = A ~ = A ~ = A, and A is both C-linked and ~,-linked. So, we have tr2v T = ~ v ~ where v#>~and~,v/J>#and A ~=(~v~)-rad(A ~.A0=(~v#)-radA = A from Lemma 2.3 of[3]. Moreover, since the lattice of uniform congruences on E(S) is modular whence semimodular, we have ~ v/z ~ r and ~ v r ~-~ and from (ii) of Lemma 2.5 we deduce that 2 v ~>-2 and 2 v T>-T.2.…”

Some remarks on modularity of the congruence lattice of regular ω-semigroups

“…Firstly, we give a characterization of regular (^-semigroups [2] Modularity of the congruence lattice 55 whose congruence lattice is strongly semimodular. Then we prove that this condition is equivalent to the modularity conditions given in [3]. Hence this paper becomes a revisit of regular o>-semigroups with modular lattice of congruences.…”

“…It is well-known that a non-simple regular w-semigroup S is either an oj-chain of groups or the disjoint union of a finite chain of groups H = [n, Hj, </> ; ] and a simple regular ^-semigroup K = K(d, K t , ^,) which is an ideal of 5. In the latter case, as in [3,4], we will use the notation S = S([n, Hj, # y ]; K, 4>) where <f> is the homomorphism which induces the retract extension of K by H°, and which is actually a homomorphism of H into K o .…”

“…In [3] and [4] the authors stated conditions under which the lattice of congruences L(5) of a regular a)-semigroup S is respectively modular or semimodular. Two other conditions, M-symmetry and strong semimodularity, are usually examined for the congruence lattice of semigroups.…”