Extensions of left regular bands by $${\mathcal {R}}$$-unipotent semigroups

Idempotent-separating extensions of regular semigroups

For a regular biordered set E, the notion of E-diagram and the associated regular semigroup was introduced in our previous paper (1995). Given a regular biordered set E, an E-diagram in a category C is a collection of objects, indexed by the elements of E and morphisms of C satisfying certain compatibility conditions. With such an E-diagram A we associate a regular semigroup RegE(A) having E as...

Extensions of Regular ‎Rings‎

Let $R$ be an associative ring with identity. An element $x in R$ is called $mathbb{Z}G$-regular (resp. strongly $mathbb{Z}G$-regular) if there exist $g in G$, $n in mathbb{Z}$ and $r in R$ such that $x^{ng}=x^{ng}rx^{ng}$ (resp. $x^{ng}=x^{(n+1)g}$). A ring $R$ is called $mathbb{Z}G$-regular (resp. strongly $mathbb{Z}G$-regular) if every element of $R$ is $mathbb{Z}G$-regular (resp. strongly $...

Extensions and Covers for Semigroups Whose Idempotents Form a Left Regular Band

Proper extensions that are “injective on L-related idempotents” of R-unipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semigroups, are described as certain subalgebras of a λ-semidirect product of a left regular band by an R-unipotent or by a glrac semigroup, respectivel...

Existentially Closed R-domains in Regular Extensions

extensions of regular ‎rings‎

let $r$ be an associative ring with identity. an element $x in r$ is called $mathbb{z}g$-regular (resp. strongly $mathbb{z}g$-regular) if there exist $g in g$, $n in mathbb{z}$ and $r in r$ such that $x^{ng}=x^{ng}rx^{ng}$ (resp. $x^{ng}=x^{(n+1)g}$). a ring $r$ is called $mathbb{z}g$-regular (resp. strongly $mathbb{z}g$-regular) if every element of $r$ is $mathbb{z}g$-regular (resp. strongly $...