Semigroups and their lattice of congruences

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

“…Thus Remark 2.6. We remark that Theorems 2.1 and 2.5 could be deduced either from Theorem 1.21 or from Theorem 5.2 of [6]. Since the proofs of these theorems have not been published, we have given here a direct proof of our result.…”

Characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero

“…Let S be an inverse semigroup whose semilattice of idempotents has finite length. The function rank ( ) a = h a a ( ) ⋅ −1 , where h a a ( ) ⋅ −1 is the height of the idempotent a a ⋅ −1 in the semilattice of idempotents of the semigroup S, is a rank function (see [6], p. 470]). In the present paper, the rank of an element is understood in exactly this sense.…”

“…In addition to [8], already cited above and dealing, in fact, with inverse ∆-semigroups, we can mention the known theorem (Theorem 6.3 in [5]) that gives necessary and sufficient conditions for a finite inverse semigroup to be permutable. Furthermore, results of Tamura and Hamilton immediately yield a theorem on the structure of a permutable Clifford semigroup [6, p. 34].…”

“…Commutative ∆-semigroups were independently described by Tamura [2] and Schein [3,4]. Most subsequent papers (for the corresponding bibliography, see [5,6] and [7]) were devoted to the clarification of the structure of ∆-semigroups that are a generalization of commutative semigroups. The structure of a finite inverse ∆-semigroup was proposed in [8].…”

“…In subsequent works, permutable semigroups close to commutative ones (e.g., duo semigroups [10], medial semigroups [11], LC -commutative semigroups [12], RDGC n -commutative semigroups [13], etc. ), permutable completely regular semigroups (see Theorems 4.8 and 4.9 in [6]), and regular ω-semigroups [14] were studied.…”