Inverse Transversal Research Articles

On Adequate Transversals

We consider adequate transversals of abundant semigroups and prove that, in a particular case, there is a natural embedding of an inverse transversal within a certain regular subsemigroup. We also introduce the concepts of simplistic, perfect, and quasi-adequate transversal and provide a number of interesting connections between these.

E-inversive Dubreil-Jacotin semigroups

Using group congruences, we obtain necessary and sufficient conditions for an ordered E-inversive semigroup to be a Dubreil-Jacotin semigroup. We also determine when such a semigroup is naturally ordered. In particular, when the subset of regular elements is a subsemigroup it contains a multiplicative inverse transversal.

On generalized inverse transversals

Let S be a regular semigroup, So an inverse subsemigroup of S. So is called a generalized inverse transversal of S, if V (x) ∩ So ≠ o. In this paper, some properties of this kind of semigroups are discussed. In particular, a construction theorem is obtained which contains some recent results in the literature as its special cases.

Identities for a Class of Regular Unary Semigroups

A class 𝒱 of regular semigroups is an e-variety if it is closed under homomorphic images, regular subsemigroups, and direct products. Let S be a regular semigroup and S° an inverse subsemigroup of S. Then S° is called an “inverse transversal of S” if it contains a unique inverse x° of each element x of S. Many important classes of regular semigroups form e-varieties of regular semigroups. However, the class of regular semigroups with inverse transversals does not form an e-variety. In this article, we consider a regular semigroup S with an inverse transversal S° as a regular unary semigroup (S, ○) with a regular unary operation “○” on S firstly. Then we prove that S is a regular semigroup with an inverse transversal S° if and only if (S, ○) satisfies the following identities (IST): Such a regular operation is called an “ist-operation,” and a regular semigroup S is called an “ist-semigroup” if there exists an ist-operation “○” on S. A regular subsemigroup T of a regular semigroup S is called an “ist-subsemigroup” if T is an ist-semigroup. A class 𝒱 of ist-semigroups is an ist-variety if it is closed under homomorphic images, ist-subsemigroups, and direct products. We characterize the set of identities of (IST) and investigate the relationship among those identities. Also, we describe the classes of regular unary semigroups which satisfy some of these identities in (IST). On the basis of that, we'll characterize the ist-varieties, in a later article.

Abundant Semigroups with Q-adequate Transversals and Some of Their Special Cases

Regular semigroups with multiplicative inverse transversals were first studied by Blyth and McFadden in 1982. In this paper, we extend the concept of inverse transversals to some kind of quasi-ideal adequate transversals of abundant semigroups. Some characterization theorems of abundant semigroups with different kinds of quasi-ideal adequate transversals are obtained. In particular, the structure theory of abundant semigroups having different kinds of quasi-ideal adequate transversals is established. Some results obtained by Blyth, McAlister and McFadden on inverse transversals in regular semigroups are hence extended to abundant semigroups.

Naturally ordered regular semigroups with an inverse monoid transversal

The notion of an inverse transversal of a regular semigroup is well-known. Here we investigate naturally ordered regular semigroups that have an inverse transversal. Such semigroups are necessarily locally inverse and the inverse transversal is a quasi-ideal. After considering various general properties that relate the imposed order to the natural order, we highlight the situation in which the inverse transversal is a monoid. The regularity of Green’s relations is also characterised. Finally, we determine the structure of a naturally ordered regular semigroup with an inverse monoid transversal.

LRT-Biordered Sets

We give a characterization and a representation of LRT-biordered sets. IT-biordered sets form a subclass of the class of all LRT-biordered sets: they are the biordered sets of idempotents of regular semigroups with an inverse transversal.

Regular Semigroups with Inverse Transversals

Let C be a semiband with an inverse transversal \(C^{\circ}\). In [7], G.T. Song and F.L. Zhu construct a fundamental regular semigroup \(T_{C,C^{\circ}}\) with an inverse transversal \(T_{C,C^{\circ}}^{\circ}\). \(T_{C,C^{\circ}}\) is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations, and the operation-although not the usual composition-is defined by means of composition. Any full regular subsemigroup T of \(T_{C,C^{\circ}}\) is a fundamental regular semigroup with inverse transversal \(T^{\circ}=T\cap T_{C,C^{\circ}}^{\circ}\). Moreover, any regular semigroup S with an inverse transversal \(S^{\circ}\) is proved to be an idempotent-separating coextension of a full regular subsemigroup T of some \(T_{C,C^{\circ}}\). By means of a full regular subsemigroup T of some \(T_{C,C^{\circ}}\) and by means of an inverse semigroup K satisfying some conditions, in this paper, we construct a regular semigroup \(T\ast K\) with inverse transversal \(T^{\circ}\ast K\) such that \(T^{\circ}\ast K\) is isomorphic to K and \((T\ast K)/\mu_{T\ast K}\) to T. Furthermore, it is proved that if S is a regular semigroup with an inverse transversal \(S^{\circ}\) then S can be constructed from the corresponding T and from \(S^{\circ}\) in this way.

Fundamental regular semigroups with inverse transversals

Let C be a regular semigroup with an inverse transversal C° and let C be generated by its idempotents. Following W. D. Munn and T. E. Hall’s idea, in this paper, a fundamental regular semigroup TC,C° with an inverse transversal TC,C°° is constructed such that the following holds. For any regular semigroup S with an inverse transversal S° and 〈E(S)〉 = C, C° = C ∩ S°, there is a homomorphism φ from S to TC,C° such that the kernel of φ is the maximum idempotent-separating congruence on S, and φ satisfies: (1) φ|C is a homomorphism from C onto 〈E(TC,C°)〉 ; (2) φ|S° is a homomorphism from S° to TC,C°°. In particular, S is fundamental if and only if S is isomorphic to a full subsemigroup of TC,C°. Our fundamental regular semigroup TC,C° is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations, and the operation—although not the usual composition—is defined by means of composition.

Biordered Sets of Regular Semigroups with Inverse Transversals1

In a regular semigroup S, an inverse subsemigroup S° of S is called an inverse transversal of S if S° contains a unique inverse x° of each element x of S. An inverse transversal S° of S is called a Q-inverse transversal of S if S° is a quasi-ideal of S.

On E-solid Semigroups with Inverse Transversals

We characterize E-solid semigroups with inverse transversals and some special such semigroups in terms of congruences. Some properties of inverse congruences over completely simple semigroups are obtained.

Amenable Orders Associated with Inverse Transversals

If S is a regular semigroup with an inverse transversal S°={x°;x∈S} then an order ≤ on S is said to be amenable with respect to S° if•(1) ≤ is compatible with the multiplication of ;•(2) on the idempotents, ≤ coincides with the natural order ≤;•(3) ≤⇒°≤°,°≤°.This notion is in fact independent of the choice of inverse transversal. Here we consider the case where S is locally inverse (equivalently, where S° is a quasi-ideal). We give a complete description of all amenable orders on S and characterise the natural order ≤n as the smallest of these. We also establish a bijection from the set of amenable orders definable on S to the set of McAlister cones of S°, whence every amenable order on S° extends to a unique amenable order on S.

Biordered Sets of Regular Semigroups with Inverse Transversals

Biordered Sets of Regular Semigroups with Inverse Transversals

ORTHODOX TRANSVERSALS OF REGULAR SEMIGROUPS

Orthodox transversals were introduced by the first author as a generalization of inverse transversals [Comm. Algebra 27(9) (1999), pp. 4275–4288]. One of our aims in this note is to consider the general case of orthodox transversals. The main results are on the sets I and Λ, two components of regular semigroups with orthodox transversals. We prove that the semibands Ī and [Formula: see text] generated by I and Λ respectively are bands. Also somewhat interesting generalizations of properties on orthodox semigroups are given; another aim in this note is to give some examples illustrating the situations that the class of regular semigroups with orthodox transversals properly includes the class of regular semigroups with inverse transversals as well as the class of orthodox semigroups.

INVERSE TRANSVERSALS ARE MUTUALLY ISOMORPHIC

It has been suspected for some time that if a regular semigroup S has an inverse transversal then all inverse transversals of S are isomorphic. Here we give a proof of this fact.

A CLASSIFICATION OF INVERSE TRANSVERSALS

A natural characterization of known types of inverse transversal leads to the introduction of several new types, which we characterize in a similar natural way. Relations between the various types produce a general classification. We also determine for which types of inverse transversal considered the congruence extension property holds.

Variants of Regular Semigroups

Let S be a regular semigroup, and let a ∈ S . Then a variant of S with respect to a is a semigroup with underlying set S and multiplication \circ defined by x \circ y = xay . In this paper, we characterise the regularity preserving elements of regular semigroups; these are the elements a such that (S,\circ) is also regular. Hickey showed that the set of regularity preserving elements can function as a replacement for the unit group when S does not have an identity. As an application, we characterise the regularity preserving elements in certain Rees matrix semigroups. We also establish connections with work of Loganathan and Chandrasekaran, and with McAlister's work on inverse transversals in locally inverse semigroups. We also investigate the structure of arbitrary variants of regular semigroups concentrating on how the local structure of a semigroup affects the structure of its variants.

Abundant Semigroups with Adequate Transversals

Suppose that S is a regular semigroup and S an inverse subsemigroup of S. S is called an inverse transversal of S, if for each a ∈ S, S meets V (a) precisely once (that is, |V (a) ∩ So| = 1). The structure theorems for regular semigroups with inverse transversals have been given by many authors (see [1, 2, 9, 10]). An analogue of an inverse transversal, which is termed an an adequate transversal, was introduced for abundant semigroups by El Qallali (see [5]). The construction for abundant semigroups satisfying the following conditions has been given in [5]: S contains a multiplicative type A transversal S; E(S) generates a regular semiband 〈E(S)〉; E(S) is a semilattice transversal of 〈E(S)〉. The purpose of this paper is to give a structure theorem for abundant semigroups with quasi-ideal adequate transversals. Many conditions considered in [5] will be removed. Hence this class of abundant semigroups properly includes the class of abundant semigroups studied in [5] (see [5, Theorem 3.9 and Theorem 4.2]). In Section 1, we collect some basic concepts and results from [3, 4, 5, 6], which are frequently used in this paper. In Section 2, we give some properties about the sets I and Λ, which are two components in the construction for this class of abundant semigroups. In their investigation into the construction for regular semigroups with inverse transversals, McAlister and McFadden proved that I and Λ are R-unipotent and L-unipotent sub-bands respectively of S (see [2, Proposition 1.7]). But the corresponding result in abundant semigroups fails to be true. We give a counterexample to show that I and Λ need not be semigroups if S is an abundant semigroup with a quasiideal adequate transversal. In Section 3, we begin by introducing the concept of (E, I,Λ)-system, then we use it to construct an abundant semigroup containing a quasi-ideal adequate transversal; conversely, every abundant semigroup with a quasiideal adequate transversal can be constructed in this way. Finally, we apply this theorem to a special case. We use the notation and terminology of [8], and assume some familiarity with the contents of [6] and [7].

Representation of congruences on regular semigroups with inverse transversals

For a regular semigroup with an inverse transversal, we have Saito’s structureW(I,S o, Λ, *, {α, β}). We represent congruences on this kind of semigroups by the so-called congruence assemblage which consist of congruences on the structure component partsI,S o and Λ. The structure of images of this type of semigroups is also presented.

Congruence lattices of regular semigroups with inverse transversals1

We deal with the congruence lattices of regular semigroups with inverse transversals. We give descriptions for the relations T ,T r T l U and V on the congruence lattice of semigroups of this kind, present expressions for the extremal values in the classes of these relations.