Restriction Semigroups Research Articles

On some problems of conjugacy in semigroups

The conjugacy relation is an important tool in the study of group theory and now has been generalized to semigroups in several methods. In particular, the p-conjugacy, o-conjugacy, c-conjugacy, i-conjugacy, n-conjugacy and r-conjugacy in semigroups are introduced and investigated extensively and some problems related to these conjugacy relations are proposed. The aim of this paper is to continue the study of conjugacy relations in semigroups around some of these problems. We first point out that p-conjugacy in a semigroup that is embeddable in a group, is not necessarily transitive. Then we obtain a simple sufficient condition under which the o-conjugacy is the universal relation and prove that this condition is also necessary in some natural classes of semigroups. Finally, we explore some properties of r-conjugacy in restriction semigroups, which extends some results on i-conjugacy in inverse semigroups. Our results have answered or partially answered some problems raised by Araújo, Kinyon, Konieczny and Malheiro in [Proc.Roy.Soc.Edinburgh Sect.A 147, 1169–1214, 2017] and [J.Algebra 533, 142–173, 2019].

On non-regular semigroups: $\lambda$-semidirect products, Zappa-Szép products and representations

This article encompasses the study of non-regular semigroups. First, we show that the $\lambda$-semidirect product $M \rtimes^{\lambda} W$ of a left $P$-Ehresmann semigroup $M$ and a left restriction semigroup $W$ is a left $P$-Ehresmann semigroup. We explore the behavior of generalized Green's relations on $M \rtimes^{\lambda} W$, and investigate some properties of $M \rtimes^{\lambda} W$. Second, the Zappa-Szép product of a right Ehresmann semigroup and its distinguished semilattice is studied. Lastly, the theory of representations of left Ehresmann semigroups with zero via homomorphisms of left Ehresmann semigroups with zero into Clifford restriction semigroups with zero is presented.

Étale categories, restriction semigroups, and their operator algebras

We define the full and reduced non-self-adjoint operator algebras associated with étale categories and restriction semigroups, answering a question posed by Kudryavtseva and Lawson in [22]. Moreover, we define the semicrossed product algebra of an étale action of a restriction semigroup on a C⁎-algebra, which turns out to be the key point when connecting the operator algebra of a restriction semigroup with the operator algebra of its associated étale category. We also prove that in the particular cases of étale groupoids and inverse semigroups our operator algebras coincide with the C⁎-algebras of the referred objects.

Proper Ehresmann semigroups

AbstractWe propose a notion of a proper Ehresmann semigroup based on a three-coordinate description of its generating elements governed by certain labelled directed graphs with additional structure. The generating elements are determined by their domain projection, range projection and σ-class, where σ denotes the minimum congruence that identifies all projections. We prove a structure result on proper Ehresmann semigroups and show that every Ehresmann semigroup has a proper cover. Our covering monoid turns out to be isomorphic to that from the work by Branco, Gomes and Gould and provides a new view of the latter. Proper Ehresmann semigroups all of whose elements admit a three-coordinate description are characterized in terms of partial multiactions of monoids on semilattices. As a consequence, we recover the two-coordinate structure result on proper restriction semigroups.

Right Clifford restriction semigroups

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 [Formula: see text]-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).

The lattice of (2, 1)-congruences on a left restriction semigroup

AbstractAll the (2, 1)-congruences on a left restriction semigroup become a complete sublattice of its lattice of congruences. The aim of this article is to study certain fundamental properties of this complete sublattice. We introducekk-,KK-,tt-, andTT-operators on this sublattice and obtain some properties. As applications, the remarkable (2, 1)-congruences are characterized. These results extend the corresponding results on inverse semigroups and ample semigroups.

Left restriction monoids from left E-completions

Given a monoid S with E any non-empty subset of its idempotents, we present a novel one-sided version of idempotent completion we call left E-completion. In general, the construction yields a one-sided variant of a small category called a constellation by Gould and Hollings. Under certain conditions, this constellation is inductive, meaning that its partial multiplication may be extended to give a left restriction semigroup, a type of unary semigroup whose unary operation models domain. We study the properties of those pairs S,E for which this happens, and characterise those left restriction semigroups that arise as such left E-completions of their submonoid of elements having domain 1. As first applications, we decompose the left restriction semigroup of partial functions on the set X and the right restriction semigroup of left total partitions on X as left and right E-completions respectively of the transformation semigroup TX on X, and decompose the left restriction semigroup of binary relations on X under demonic composition as a left E-completion of the left-total binary relations. In many cases, including these three examples, the construction embeds in a semigroup Zappa-Szép product.

Ordered Ehresmann semigroups and categories

Ehresmann semigroups may be viewed as biunary semigroups equipped with domain and range operations satisfying some equational laws. Motivated by some of the main examples, we here define ordered Ehresmann semigroups, and consider their basic properties as well as special cases in which the order is algebraically definable. In particular, one and two-sided restriction semigroups equipped with their natural orders are characterized within the class of ordered Ehresmann semigroups. The main result is an ESN-style theorem for ordered Ehresmann semigroups with particular reference to the special cases.

On left restriction semigroups with zero

In this article, we give the notion of left restriction meet-semigroup, and establish some results regarding atomistic left restriction semigroups. Then we discuss decompositions of (non-zero) semigroups with zero by proving a decomposition theorem. We also show that every atomistic left restriction semigroup S can be decomposed as an orthogonal sum of atomistic left restriction semigroups Ni, where each summand Ni is an irreducible ideal of S. Finally, properties of the summands Ni, when S embeds in some PT X the partial transformation monoid on a set X, are investigated.

On orthodox P-restriction semigroups

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

Congruences on glrac semigroups (I)

The theory of congruences on semigroups is an important part in the theory of semigroups. The aim of this paper is to study (2,1)-congruences on a glrac semigroup. It is proved that the (2,1)-congruences on a glrac semigroup become a complete sublattice of its lattice of congruences. Especially, the structures of left restriction semigroup (2,1)-congruences and the projection-separating (2,1)-congruences on a glrac semigroup are established. Also, we demonstrate that they are both complete sublattice of (2,1)-congruences and consider their relations with respect to complete lattice homomorphism.

Partial actions and proper extensions of two-sided restriction semigroups

We prove a structure result on proper extensions of two-sided restriction semigroups in terms of partial actions, generalizing respective results for monoids and for inverse semigroups and upgrading the latter. We introduce and study several classes of partial actions of two-sided restriction semigroups that generalize partial actions of monoids and of inverse semigroups. We establish an adjunction between the category P ( S ) of proper extensions of a restriction semigroup S and a category A ( S ) of partial actions of S subject to certain conditions going back to the work of O'Carroll. In the category A ( S ) , we specify two isomorphic subcategories , one being reflective and the other one coreflective, each of which is equivalent to the category P ( S ) .

Some algebraic structures on the generalization general products of monoids and semigroups

For arbitrary monoids A and B, in Cevik et al. (Hacet J Math Stat 2019:1–11, 2019), it has been recently defined an extended version of the general product under the name of a higher version of Zappa products for monoids (or generalized general product) A^{oplus B}_{delta }bowtie _{psi }B^{oplus A} and has been introduced an implicit presentation as well as some theories in terms of finite and infinite cases for this product. The goals of this paper are to present some algebraic structures such as regularity, inverse property, Green’s relations over this new generalization, and to investigate some other properties and the product obtained by a left restriction semigroup and a semilattice.

How to generalise demonic composition

Demonic composition is defined on the set of binary relations over the non-empty set X, \(Rel_X\), and is a variant of standard or “angelic” composition. It arises naturally in the setting of the theory of non-deterministic computer programs, and shares many of the nice features of ordinary composition (it is associative, and generalises composition of functions). When equipped with the operations of demonic composition and domain, \(Rel_X\) is a left restriction semigroup (like \(PT_X\), the semigroup of partial functions on X), whereas usual composition and domain give a unary semigroup satisfying weaker laws. By viewing \(Rel_X\) under a restricted version of its usual composition and domain as a constellation (a kind of “one-sided” category), we show how this demonic left restriction semigroup structure arises on \(Rel_X\), placing it in a more general context. The construction applies to any unary semigroup with a “domain-like” operation satisfying certain minimal conditions which we identify. In particular it is shown that using the construction, any Baer \(*\)-semigroup S can be given a left restriction semigroup structure which is even an inverse semigroup if S is \(*\)-regular. It follows that the semigroup of \(n\times n\) matrices over the real or complex numbers is an inverse semigroup with respect to a modified notion of product that almost always agrees with the usual matrix product, and in which inverse is pseudoinverse (Moore–Penrose inverse).

Completions of Generalized Restriction P-Restriction Semigroups

Generalized restriction P-restriction semigroups are common generalizations of restriction semigroups and generalized inverse $$*$$ -semigroups. Gomes and Szendrei (resp. Ohta and Imaoka) have shown that every restriction semigroup (every generalized inverse $$*$$ -semigroup) can be embedded in a complete, infinitely distributive restriction semigroup (resp. a $$*$$ -complete, infinitely distributive generalized inverse $$*$$ -semigroup). The main aim of this paper is to obtain an entirely corresponding result for generalized restriction P-restriction semigroups. Specifically, among other things, we show that every generalized restriction P-restriction semigroup can be (2,1,1)-embedded in a complete, infinitely distributive generalized restriction P-restriction semigroup. Our results generalize and enrich the corresponding results of Gomes, Szendrei, Ohta and Imaoka.

Computing the congruence class of some left restriction semigroups in ℘𝕴_X

In this paper, we present some examples of left restriction semigroups embedded in the partial transformation ℘𝕴_X. Also, we compute and enumerates the congruence-class ~R and the semilattice of idempotents E_X inherent in the examples generated.

Pseudo-amenability and pseudo-contractibility of restricted semigroup algebra

Abstract In this article the pseudo-amenability and pseudo-contractibility of restricted semigroup algebra l r 1 ( S ) $l_r^1(S)$ and semigroup algebra, l 1(Sr) on restricted semigroup, Sr are investigated for different classes of inverse semi-groups such as Brandt semigroup, and Clifford semigroup. We particularly show the equivalence between pseudo-amenability and character amenability of restricted semigroup algebra on a Clifford semigroup and semigroup algebra on a restricted semigroup. Moreover, we show that when S = M 0(G, I)is a Brandt semigroup, pseudo-amenability of l 1(Sr) is equivalent to its pseudo-contractibility.

Restriction $$\omega $$ ω -semigroups

The purpose of this paper is to investigate restriction $$\omega $$ -semigroups. Here a restriction $$\omega $$ -semigroup is a generalisation of an inverse $$\omega $$ -semigroup. We give a description of a class of restriction $$\omega $$ -semigroups, namely, restriction $$\omega $$ -semigroups with an inverse skeleton. We show that a restriction $$\omega $$ -semigroup with an inverse skeleton is an ideal extension of a $$\widetilde{\mathcal {J}}$$ -simple restriction $$\omega $$ -semigroup by a restriction semigroup with a finite chain of projections with a zero adjoined. This result is analogous to Munn’s result for inverse $$\omega $$ -semigroups. In addition, we show that the Bruck–Reilly semigroup of a strong semilattice of monoids indexed by a finite chain is a $$\widetilde{\mathcal {J}}$$ -simple restriction $$\omega $$ -semigroup with an inverse skeleton, conversely, every $$\widetilde{\mathcal {J}}$$ -simple restriction $$\omega $$ -semigroup with an inverse skeleton arises in this way.

ON PSEUDO-EHRESMANN SEMIGROUPS

As generalizations of inverse semigroups, Ehresmann semigroups are introduced by Lawson and investigated by many authors extensively in the literature. In particular, Lawson has proved that the category of Ehresmann semigroups and admissible morphisms is isomorphic to the category of Ehresmann categories and strongly ordered functors, which generalizes the well-known Ehresmann–Schein–Nambooripad (ESN) theorem for inverse semigroups. From a varietal perspective, Ehresmann semigroups are derived from reducts of inverse semigroups. In this paper, inspired by the approach of Jones [‘A common framework for restriction semigroups and regular $\ast$-semigroups’, J. Pure Appl. Algebra216 (2012), 618–632], Ehresmann semigroups are extended from a varietal perspective to pseudo-Ehresmann semigroups derived instead from reducts of regular semigroups with a multiplicative inverse transversal. Furthermore, motivated by the method used by Gould and Wang [‘Beyond orthodox semigroups’, J. Algebra368 (2012), 209–230], we introduce the notion of inductive pseudocategories over admissible quadruples by which pseudo-Ehresmann semigroups are described. More precisely, we show that the category of pseudo-Ehresmann semigroups and (2,1,1,1)-morphisms is isomorphic to the category of inductive pseudocategories over admissible quadruples and pseudofunctors. Our work not only generalizes the result of Lawson for Ehresmann semigroups but also produces a new approach to characterize regular semigroups with a multiplicative inverse transversal.

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.