Additively orthodox semirings with special transversals

<abstract><p>A semiring $ (S, +, \cdot) $ is called additively orthodox semiring if its additive reduct $ (S, +) $ is a orthodox semigroup. In this paper, by introducing some special semiring transversals as the tools, the constructions of additively orthodox semirings with a skew-ring transversal or with a generalized Clifford semiring transversal are established. Meanwhile, it is shown that an additively orthodox semiring with a generalized Clifford semiring transversal is a b-lattice of additively orthodox semirings with skew-ring transversals. Consequently, the corresponding results of Clifford semirings and generalized Clifford semirings in reference (M. K. Sen, S. K. Maity, K. P. Shum, Clifford semirings and generalized Clifford semirings, Taiwan. J. Math., 9 (2005), 433–444) and completely regular semirings in reference (S. K. Maity, M. K. Sen, K. P. Shum, On completely regular semirings, Bull. Cal. Math. Soc., 98 (2006), 319–328) are extended and strengthened.</p></abstract>

On E-semiabundant semigroups with a multiplicative restriction transversal

Multiplicative inverse transversals of regular semigroups were introduced by Blyth and McFadden in 1982. Since then, regular semigroups with an inverse transversal and their generalizations, such as regular semigroups with an orthodox transversal and abundant semigroups with an ample transversal, are investigated extensively in literature. On the other hand, restriction semigroups are generalizations of inverse semigroups in the class of non-regular semigroups. In this paper we initiate the investigations of E-semiabundant semigroups by using the ideal of "transversals". More precisely, we first introduce multiplicative restriction transversals for E-semiabundant semigroups and obtain some basic properties of E-semiabundant semigroups containing a multiplicative restriction transver- sal. Then we provide a construction method for E-semiabundant semigroups containing a multiplicative restriction transversal by using the Munn semigroup of an admissible quadruple and a restriction semigroup under some natural conditions. Our construction is similar to Hall's spined product construction of an orthodox semigroup. As a corollary, we obtain a new construction of a regular semigroup with a multiplicative inverse transversal and an abundant semigroup having a multiplicative ample transversal, which enriches the corresponding results obtained by Blyth-McFadden and El-Qallali, respectively.

On endomorphisms of power-semigroups

An involuted semilattice [Formula: see text] is a semilattice [Formula: see text] with an identity element [Formula: see text] and with an involution [Formula: see text] satisfying [Formula: see text] and [Formula: see text]. We consider involuted semilattices [Formula: see text] with an identity [Formula: see text] such that there is a subsemilattice [Formula: see text] without [Formula: see text] with the property that any [Formula: see text] belongs to exactly one of the following four sets : [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. In this paper, we introduce an associative binary operation [Formula: see text] on [Formula: see text] in the following quite natural way: [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text] and characterize all endomorphisms of the orthodox semigroup [Formula: see text].

End-regular and End-orthodox generalized lexicographic products of bipartite graphs

A graph X is said to be End-regular (End-orthodox) if its endomorphism monoid End(X) is a regular (orthodox) semigroup. In this paper, we determine the End-regular and the End-orthodox generalized lexicographic products of bipartite graphs.

Certain congruences on eventually regular semigroups I

A semigroup is called eventually regular if each of its elements has a regular power. In this paper we study certain fundamental congruences on an eventually regular semigroup. We generalize some results of Howie and Lallement (1966) and LaTorre (1983). In particular, we give a full description of the semilattice of group congruences (together with the least such a congruence) on an arbitrary eventually regular (orthodox) semigroup. Moreover, we investigate UBG-congruences on an eventually regular semigroup. Finally, we study the eventually regular subdirect products of an E-unitary semigroup and a Clifford semigroup.

On Epiorthodox Semigroups

It has been well known that the band of idempotents of a naturally ordered orthodox semigroup satisfying the “strong Dubreil-Jacotin condition” forms a normal band. In the literature, the naturally ordered orthodox semigroups satisfying the strong Dubreil-Jacotin condition were first considered by Blyth and Almeida Santos in 1992. Based on the name “epigroup” in the paper of Blyth and Almeida Santos and also the name “epigroups” proposed by Shevrin in 1955; we now call the naturally ordered orthodox semigroups satisfying the Dubreil-Jacotin condition theepiorthodox semigroups. Because the structure of this kind of orthodox semigroups has not yet been described, we therefore give a structure theorem for the epi-orthodox semigroups.

On quasi-F-orthodox semigroups

An orthodox semigroup S is termed quasi-F-orthodox if the greatest inverse semigroup homomorphic image of S1 is F-inverse. In this paper we show that each quasi-F-orthodox semigroup is embeddable into a semidirect product of a band by a group. Furthermore, we present a subclass in the class of quasi-F-orthodox semigroups whose members S are embeddable into a semidirect product of a band B by a group in such a way that B belongs to the band variety generated by the band of idempotents in S. In particular, this subclass contains the F-orthodox semigroups and the idempotent pure homomorphic images of the bifree orthodox semigroups.

Extensions of regular orthogroups by groups

A common generalization of the author's embedding theorem concerning the E-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.

EXTENSIONS OF REGULAR ORTHOGROUPS BY INVERSE SEMIGROUPS

Let V be a variety of regular orthogroups, i.e. completely regular orthodox semigroups whose band of idempotents is regular. Let S be an orthodox semigroup which is a (normal) extension of an orthogroup K from V by an inverse semigroup G, that is, there is a congruence ρ on S such that the semigroup ker ρ of all idempotent related elements of S is isomorphic to K and S/ρ≅G. It is shown that S can be embedded into an orthodox subsemigroup T of a double semidirect product A**G where A belongs to V. Moreover T itself can be chosen to be an extension of a member from V by G. If in addition ρ is a group congruence we obtain a recent result due to M.B. Szendrei [16] which says that each orthodox semigroup which is an extension of a regular orthogroup K by a group G can be embedded into a semidirect product of an orthogroup K′ by G where K′ belongs to the variety of orthogroups generated by K.