Regular Semigroups Research Articles

New Properties of Anti Fuzzy Ideals of Regular Semigroups

In this article, we study some properties of anti-fuzzy sub-semigroup, anti fuzzy left (right, two sided) ideal, anti fuzzy ideal, anti fuzzy generalized bi-ideal, anti fuzzy interior ideals and anti fuzzy two sided ideal of regular semigroup. Also, we characterized regular LA-semigroup in terms of their anti fuzzy ideal.

Structure of Strict Abundant Semigroups

We introduce a new class of semigroups called strict abundant semigroups, which are concordant semigroups and subdirect products of completely [Formula: see text]-simple abundant semigroups and completely 0-[Formula: see text]-simple primitive abundant semigroups. A general construction and a tree structure of such semigroups are established. Consequently, the corresponding structure theorems for strict regular semigroups given by Auinger in 1992 and by Grillet in 1995 are generalized and extended. Finally, an example of strict abundant semigroups is also given.

Left and right regular elements of the semigroups of transformations preserving an equivalence relation and a cross-section

Let [Formula: see text] be the semigroup of all transformations on a set [Formula: see text]. For an arbitrary equivalence relation [Formula: see text] on [Formula: see text] and a cross-section [Formula: see text] of the partition [Formula: see text] induced by [Formula: see text], let [Formula: see text] [Formula: see text] Then [Formula: see text] and [Formula: see text] are subsemigroups of [Formula: see text]. In this paper, we characterize left regular, right regular and completely regular elements of [Formula: see text] and [Formula: see text]. We also investigate conditions for which of these semigroups to be left regular, right regular and completely regular semigroups.

English

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Regularity in the Semigroup of Transformations Preserving a Zig-Zag Order

It is well known that the semigroup T(X) of all transformations on a set X is regular, but its subsemigroups do not need to be. Consider an ordered set $$(X;\le )$$ whose order forms a path with alternating orientation. The semigroup $$\mathrm{OT}(X)$$ of all order-preserving transformations on $$(X;\le )$$ is a subsemigroup of T(X). Some characterizations of regular semigroups $$\mathrm{OT}(X)$$ were given. For a non-empty subset S of X, we define the set $$\begin{aligned} \mathrm{OT}_S(X)=\{\alpha \in \mathrm{OT}(X)\mid {{\,\mathrm{ran}\,}}\alpha =S \}. \end{aligned}$$In this paper, we show that in general, $$\mathrm{OT}_S(X)$$ is not a subsemigroup of T(X). A necessary and sufficient condition for $$\mathrm{OT}_S(X)$$ to be a regular subsemigroup of T(X) is given. Some regular subsemigroups of T(X) that are contained in $$\mathrm{OT}_S(X)$$ are studied. We investigate the left (right) regularity of $$\mathrm{OT}(X)$$. We obtain that left regular and right regular elements in $$\mathrm{OT}(X)$$ are identical. A characterization of left (right) elements in $$\mathrm{OT}(X)$$ is given. We classify all left (right) regular semigroups $$\mathrm{OT}(X)$$.

A conjecture for varieties of completely regular semigroups

Abstract The class 𝒞ℛ of completely regular semigroups considered with the unary operation of inversion within maximal subgroups forms a variety. The B-relation on the lattice ℒ(𝒞ℛ) of subvarieties of 𝒞ℛ identifies two varieties if they contain the same bands. Its classes are intervals with the set Δ of upper ends of these intervals. Canonical varieties form part of Δ. Previously we determined the sublattice Ψ of ℒ(𝒞ℛ) generated by the variety 𝒞𝒮 of completely simple semigroups and six canonical varieties. The conjecture is that the sublattice of ℒ(𝒞ℛ) generated by 𝒞𝒮 and canonical varieties follows the pattern of the structure of Ψ.

Flows on Classes of Regular Semigroups and Cauchy Categories

We consider the structure of the flow monoid for some classes of regular semigroups (which are special case of flows on categories) and for Cauchy categories. In detail, we characterize flows for Rees matrix semigroups, rectangular bands, and full transformation semigroups and also describe the Cauchy categories for some classes of regular semigroups such as completely simple semigroups, Brandt semigroups, and rectangular bands. In fact, we obtain a general structure for the flow monoids on Cauchy categories.

Correction to: Pseudovarieties of Ordered Completely Regular Semigroups

Correction to: Pseudovarieties of Ordered Completely Regular Semigroups

On the lattice of biorder ideals of regular rings

The study of biordered set plays a significant role in describing the structure of a regular semigroup and since the definition of regularity involves only the multiplication in the ring, it is natural that the study of semigroups plays a significant role in the study of regular rings. Here, we extend the biordered set approach to study the structure of the regular semigroup [Formula: see text] of a regular ring [Formula: see text] by studying the idempotents [Formula: see text] of the regular ring and show that the principal biorder ideals of the regular ring [Formula: see text] form a complemented modular lattice and certain properties of this lattice are studied.

Involution matchings, the semigroup of orientation-preserving and orientation-reversing mappings, and inverse covers of the full transformation semigroup

We continue the study of permutations of a finite regular semigroup that map each element to one of its inverses, providing a complete description in the case of semigroups whose idempotent generated subsemigroup is a union of groups. We show, in two ways, how to construct an involution matching on the semigroup of all transformations which either preserve or reverse orientation of a cycle. Finally, as an application, we use involution matchings to prove that when the base set has at least four members, a finite full transformation semigroup has no cover by inverse subsemigroups that is closed under intersection.

Pseudovarieties of Ordered Completely Regular Semigroups

This paper is a contribution to the theory of finite semigroups and their classification in pseudovarieties, which is motivated by its connections with computer science. The question addressed is what role can play the consideration of an order compatible with the semigroup operation. In the case of unions of groups, so-called completely regular semigroups, the problem of which new pseudovarieties appear in the ordered context is solved. As applications, it is shown that the lattice of pseudovarieties of ordered completely regular semigroups is modular and that taking the intersection with the pseudovariety of bands defines a complete endomorphism of the lattice of all pseudovarieties of ordered semigroups.

Semigroups with finitely generated universal left congruence

We consider semigroups such that the universal left congruence omega ^{ell } is finitely generated. Certainly a left noetherian semigroup, that is, one in which all left congruences are finitely generated, satisfies our condition. In the case of a monoid the condition that omega ^{ell } is finitely generated is equivalent to a number of pre-existing notions. In particular, a monoid satisfies the homological finiteness condition of being of type left-{text {FP}}_1 exactly when omega ^{ell } is finitely generated.Our investigations enable us to classify those semigroups such that omega ^{ell } is finitely generated that lie in certain important classes, such as strong semilattices of semigroups, inverse semigroups, Rees matrix semigroups (over semigroups) and completely regular semigroups. We consider closure properties for the class of semigroups such that omega ^{ell } is finitely generated, including under morphic image, direct product, semi-direct product, free product and 0-direct union. Our work was inspired by the stronger condition, stated for monoids in the work of White, of being pseudo-finite. Where appropriate, we specialise our investigations to pseudo-finite semigroups and monoids. In particular, we answer a question of Dales and White concerning the nature of pseudo-finite monoids.

Some special congruences on completely regular semigroups

This paper enriches the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower t and lower k. Three classes of completely regular semigroups, namely semigroups for which is a cryptogroup, semigroups for which is a cryptogroup and semigroups for which κ is over rectangular bands, are studied. and are found to be the least congruences on S such that the quotient semigroups are semigroups for which is a cryptogroup, is a cryptogroup and κ is over rectangular bands, respectively. The results obtained present a response to three problems in Petrich and Reilly’s textbook “Completely Regular Semigroups”.

Characterizations of regular semi-groups by interval-valued $Q$-fuzzy ideals with thresholds $(overline{alpha},overline{eta})$

Characterizations of regular semi-groups by interval-valued $Q$-fuzzy ideals with thresholds $(overline{alpha},overline{eta})$

Ordered regular semigroups with biggest associates

Ordered regular semigroups with biggest associates

Construction of Inverse Unit Regular Monoids from a Semilattice and a Group

This paper is a continuation of a previous paper [6] in which the structure of certain unit regular semigroups called R-strongly unit regular monoids has been studied. A monoid S is said to be unit regular if for each element s Î S there exists an element u in the group of units G of S such that s = sus. Hence where su is an idempotent and is a unit. A unit regular monoid S is said to be a unit regular inverse monoid if the set of idempotents of S form a semilattice. Since inverse monoids are R unipotent, every element of a unit regular inverse monoid can be written as s = eu where the idempotent part e is unique and u is a unit. Here we give a detailed study of inverse unit regular monoids and the results are mainly based on [10]. The relations between the semilattice of idempotents and the group of units in unit regular inverse monoids are better identified in this case. .

Inductive groupoids and cross-connections of regular semigroups

There are two major structure theorems for an arbitrary regular semigroup using categories, both due to Nambooripad. The first construction using inductive groupoids departs from the biordered set structure of a given regular semigroup. This approach belongs to the realm of the celebrated Ehresmann--Schein--Nambooripad Theorem and its subsequent generalisations. The second construction is a generalisation of Grillet's work on cross-connected partially ordered sets, arising from the principal ideals of the given semigroup. In this article, we establish a direct equivalence between these two seemingly different constructions. We show how the cross-connection representation of a regular semigroup may be constructed directly from the inductive groupoid of the semigroup, and vice versa.

Cross-connections and variants of the full transformation semigroup

Cross-connection theory propounded by K. S. S. Nambooripad describes the ideal structure of a regular semigroup using the categories of principal left (right) ideals. A variant $\mathscr{T}_X^\theta$ of the full transformation semigroup $(\mathscr{T}_X,\cdot)$ for an arbitrary $\theta \in \mathscr{T}_X$ is the semigroup $\mathscr{T}_X^\theta= (\mathscr{T}_X,\ast)$ with the binary operation $\alpha \ast \beta = \alpha\cdot\theta\cdot\beta$ where $\alpha, \beta \in \mathscr{T}_X$. In this article, we describe the ideal structure of the regular part $Reg(\mathscr{T}_X^\theta)$ of the variant of the full transformation semigroup using cross-connections. We characterize the constituent categories of $Reg(\mathscr{T}_X^\theta)$ and describe how they are \emph{cross-connected} by a functor induced by the sandwich transformation $\theta$. This lead us to a structure theorem for the semigroup and give the representation of $Reg(\mathscr{T}_X^\theta)$ as a cross-connection semigroup. Using this, we give a description of the biordered set and the sandwich sets of the semigroup.

Characterizations of Right Weakly Regular Semigroups in Terms of Generalized Cubic Soft Sets

Cubic sets are the very useful generalization of fuzzy sets where one is allowed to extend the output through a subinterval of [ 0 , 1 ] and a number from [ 0 , 1 ] . Generalized cubic sets generalized the cubic sets with the help of cubic point. On the other hand Soft sets were proved to be very effective tool for handling imprecision. Semigroups are the associative structures have many applications in the theory of Automata. In this paper we blend the idea of cubic sets, generalized cubic sets and semigroups with the soft sets in order to develop a generalized approach namely generalized cubic soft sets in semigroups. As the ideal theory play a fundamental role in algebraic structures through this we can make a quotient structures. So we apply the idea of neutrosophic cubic soft sets in a very particular class of semigroups namely weakly regular semigroups and characterize it through different types of ideals. By using generalized cubic soft sets we define different types of generalized cubic soft ideals in semigroups through three different ways. We discuss a relationship between the generalized cubic soft ideals and characteristic functions and cubic level sets after providing some basic operations. We discuss two different lattice structures in semigroups and show that in the case when a semigroup is regular both structures coincides with each other. We characterize right weakly regular semigroups using different types of generalized cubic soft ideals. In this characterization we use some classical results as without them we cannot prove the inter relationship between a weakly regular semigroups and generalized cubic soft ideals. This generalization leads us to a new research direction in algebraic structures and in decision making theory.

A note on additively completely regular seminearrings

In the endeavour of obtaining semigroup theoretic analogues i.e., the analogues of structure theorems of completely regular semigroups in the setting of additively regular seminearrings we could obtain some results in Mukherjee et al. (Commun Algebra 45(12):5111–5122, 2017). But we could not obtain the analogue of (i) ‘A semigroup is Clifford if and only if it is strong semilattice of groups’ and (ii) ‘ A semigroup is completely regular if and only if it is a union of groups’. In Mukherjee et al. (Commun Algebra, https://doi.org/10.1080/00927872.2018.1524011) we could obtain the analogue of (i) for some restricted type of left (right) Clifford seminearrings. The main purpose of this paper is to complete the remaining task i.e., to obtain the analogue of (ii) for a class of additively completely regular seminearrings. In order to accomplish this we have characterized those seminearrings which are union of near-rings (zero-symmetric near-rings) in the class of additively completely regular seminearrings.