Extensions of regular orthogroups by groups

Abstract

A common generalization of the author's embedding theorem concerning the £-unitary regular semigroups with regular band of idempotents, and Billhardt's and Ismaeel's embedding theorem on the inverse semigroups, the closure of whose set of idempotents is a Clifford semigroup, is presented. We prove that each orthodox semigroup with a regular band of idempotents, which is an extension of an orthogroup K by a group, can be embedded into a semidirect product of an orthogroup K' by a group, where K' belongs to the variety of orthogroups generated by K. The proof is based on a criterion of embeddability into a semidirect product of an orthodox semigroup by a group and uses bilocality of orthogroup bivarieties.

Structure theory of regular semigroups

“…Theorem 5.7 (Szendrei [61]). Every extension of a regular orthogroup K by a group G is embeddable in a semidirect product of a regular orthogroup K ′ by G where K ′ belongs to the variety of orthogroups generated by K.…”

Extensions by inverse semigroups andλ -semidirect products

“…Furthermore, this approach was used in [13] and [3], and the same ideas also appeared in [8] in connection with regular extensions of Cli ord semigroups by groups. The canonical embedding approach was generalized to the case of regular extensions of orthodox semigroups by groups in [14] and was applied to regular extensions of regular orthogroups by groups in [15]. Although the roots of the investigations of regular extensions by inverse semigroups go back to L. O'Carroll [10] and C. H. Houghton [6], the newer results on regular extensions by groups directed attention to the problem of whether similar results could be achieved for extensions by inverse semigroups.…”