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Right C-restriction semigroups are the common generalizations of right C-rpp semigroups and right C-lpp semigroups. The structure of such a semigroup is established in terms of the Δ -product of a right regular band and a C-restriction semigroup. As its applications, we give the structures of right C-rpp semigroups and right C-lpp semigroups. Also, we give the dual wreath product structure of right C-restriction semigroups. These results extend the related results in (Semigroup Forum 68 (2004) 280–292) and (Algebra Colloquium 14 (2007) 285–294). [ABSTRACT FROM AUTHOR]

The investigation of orthodox P -restriction semigroups was initiated by Jones in 2014 as generalizations of orthodox * -semigroups. The aim of this paper is to further study orthodox P -restriction semigroups based on the known results of Jones. After establishing a construction theorem for orthodox P -restriction semigroups, we introduce proper P -restriction semigroups (which are necessarily orthodox) and prove that every (finite) orthodox P -restriction semigroup has a (finite) proper cover. Our results enrich and extend existing results for restriction semigroups and orthodox * -semigroups. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce and study quasi-order Γ -semihypergroups that is a generalization of quasi-order semihypergroups. Also, we introduce quasi-order semihypergroups associated with quasi-order Γ -semihypergroups and some important results obtained. Moreover, we study complete parts and fundamental relation on quasi-order Γ -semihypergroups and prove that the fundamental relation is transitive. [ABSTRACT FROM AUTHOR]

An adequate semigroup S is said to be ample if for any e2 = e, a ∈ S, ae = (ae)†a and ea = a(ea)*. It is well known that inverse semigroups are ample semigroups. The purpose of this paper is to study matrix representations of an ample semigroup. Some properties of ample semigroups are obtained. It is proved that any indecomposable good matrix representations of an ample semigroup can be constructed by using those of weak Brandt semigroups. [ABSTRACT FROM AUTHOR]

It was first proved by McAlister in 1983 that every locally inverse semigroup is a locally isomorphic image of a regular Rees matrix semigroup over an inverse semigroup and Lawson in 2000 further generalized this result to some special locally adequate semigroups. In this paper, we show how these results can be extended to a class of locally Ehresmann semigroups. [ABSTRACT FROM AUTHOR]

Abundant semigroups expressed as a perfect rectangular band of adequate semigroups will be investigated. A known theorem of Pastijn–Petrich in 1984 on perfect rectangular bands of inverse semigroups will be generalized from the class of regular semigroups to the class of abundant semigroups. A Rees matrix representation of such abundant semigroups will be established and some characterization theorems will be consequently given. [ABSTRACT FROM AUTHOR]