Quasi-ideal Ehresmann transversals: The spined product structure

In any U-abundant semigroup with an Ehresmann transversal, two significant components R and L are introduced in this paper and described by Green’s ∼ \sim -relations. Some interesting properties associated with R and L are explored and some equivalent conditions for the Ehresmann transversal to be a quasi-ideal are acquired. Finally, a spined product structure theorem is established for a U-abundant semigroup with a quasi-ideal Ehresmann transversal by means of R and L.

On refined generalised quasi-adequate transversals

Some properties and characterizations for abundant semigroups with generalised quasiadequate transversals are explored. In such semigroups, an interesting property [?a,b ? Re1S, VSo(a) ? VSo (b) ? 0 ? VSo (a) = VSo (b)] is investigated and thus the concept of refined generalised quasi-adequate transversals, for short, RGQA transversals is introduced. It is shown that RGQA transversals are the real common generalisations of both orthodox transversals and adequate transversals in the abundant case. Finally, by means of two abundant semigroups R and L, a spined product structure theorem for an abundant semigroup with a quasi-ideal RGQA transversal is established.

Semiprimeness of semigroup algebras

Abundant semigroups originate from p.p. rings and are generalizations of regular semigroups. The main aim of this paper is to study the primeness and the primitivity of abundant semigroup algebras. We introduce and study D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices of a semigroup. Based on D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices, we determine when a contracted semigroup algebra of a primitive abundant semigroup is prime (respectively, semiprime, semiprimitive, or primitive). It is well known that for any algebra A {\mathcal{A}} with unity, A {\mathcal{A}} is primitive (prime) if and only if so is M n ( A ) {M}_{n}\left({\mathcal{A}}) . Our results can be viewed as some kind of generalizations of such a known result. In addition, it is proved that any contracted semigroup algebra of a locally ample semigroup whose set of idempotents is locally finite (respectively, locally pseudofinite and satisfying the regularity condition) is isomorphic to some contracted semigroup algebra of primitive abundant semigroups. Moreover, we obtain sufficient and necessary conditions for these classes of contracted semigroup algebras to be prime (respectively, semiprime, semiprimitive, or primitive). Finally, the structure of simple contracted semigroup algebras of idempotent-connected abundant semigroups is established. Our results enrich and extend the related results on regular semigroup algebras.

A ∗-prehomomorphism of a type B semigroup

Type B semigroups are generalizations of inverse semigroups, and every inverse semigroup admits an [Formula: see text]-unitary cover (M. Petrich, Inverse Semigroups (Wiley, New York, 1984)). Motivated by studying [Formula: see text]-unitary cover for inverse semigroups, and as a continuation of Petrich’s works in inverse semigroups, in this paper, we first introduce the concept of ∗-prehomomorphism of a type B semigroup. After obtaining some basic properties, we get some structure theorems and give some conditions for a type B semigroup which is constructed by using the ∗-prehomomorphism to be proper. In particular, we introduce the notion of [Formula: see text]-unitary good cover for an abundant semigroup, and prove that every type B semigroup with compatible natural partial order admits an [Formula: see text]-unitary good cover.

Self-injectivity of semigroup algebras

It is proved that for an IC abundant semigroup (a primitive abundant semigroup; a primitively semisimple semigroup) S and a field K, if K 0[S] is right (left) self-injective, then S is a finite regular semigroup. This extends and enriches the related results of Okniński on self-injective algebras of regular semigroups, and affirmatively answers Okniński’s problem: does that a semigroup algebra K[S] is a right (respectively, left) self-injective imply that S is finite? (Semigroup Algebras, Marcel Dekker, 1990), for IC abundant semigroups (primitively semisimple semigroups; primitive abundant semigroups). Moreover, we determine the structure of K 0[S] being right (left) self-injective when K 0[S] has a unity. As their applications, we determine some sufficient and necessary conditions for the algebra of an IC abundant semigroup (a primitively semisimple semigroup; a primitive abundant semigroup) over a field to be semisimple.

On abundant semigroups withweak normal idempotents

A weak normal idempotent of an abundant semigroup was introduced by Guo [7]. In this paper, weak normal idempotents and normal idempotents of abundant semigroups are respectively characterized in many different ways. These results enable us to obtain an example which shows that the class of normal idempotents of abundant semigroups is a proper subclass of normal idempotents of abundant semigroups. Furthermore, this example tell us that there exists a non-regular abundant semigroup containing a weak normal idempotent. At last, we investigate the relationships between weak normal idempotents and normal idempotents and deduce that the main result of [2] can not be generalized into the class of abundant semigroups.

Cross-connection structure of concordant semigroups

Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalized Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the category of concordant semigroups and the category of cross-connected consistent categories. In the process, we illustrate how our construction is a generalization of the cross-connection analysis of regular semigroups. We also identify the inductive cancellative category associated with a pair of cross-connected consistent categories.

Good congruences on abundant semigroups with quasi-ideal adequate transversals

The aim of this paper is to study the congruences on abundant semigroups with quasi-ideal adequate transversals. The good congruences on an abundant semigroup with a quasi-ideal adequate transversal S° are described by the equivalence triple abstractly which consists of equivalences on the structure component parts I, S° and Λ. Also, it is shown that the set of all good congruences on this kind of semigroup forms a complete lattice.

The product of quasi-ideal refined generalised quasi-adequate transversals

As the real common generalisations of both orthodox transversals and adequate transversals in the abundant case, the concept of refined generalised quasi-adequate transversal, for short, RGQA transversal was introduced by Kong and Wang. In this paper, an interesting characterization for a generalised quasi-adequate transversal to be refined is acquired. It is shown that the product of every two quasi-ideal RGQA transversals of the abundant semigroup S satisfying the regularity condition is a quasi-ideal RGQA transversal of S and that all quasi-ideal RGQA transversals of S compose a rectangular band. The related results concerning adequate transversals are generalised and enriched.