Orders in Strict Regular Semigroups

 A subsemigroup S of a semigroup Q is an order in Q if for every there exist such that , where a and d are contained in (maximal) subgroups of Q, and and are their inverses in these subgroups. A regular semigroup S is strict if it is a subdirect product of completely (0-)simple semigroups. We construct all orders and involutions in Auinger’s model of a strict regular semigroup. This is used to find necessary and sufficient conditions on an involution on an order S in a strict regular semigroup Q for extendibility to an involution on Q. (Received 27 April 1999; in revised form 20 October 1999)     

Weakly Strict Regular Semigroups

Weakly strict regular semigroups WS represent a generalization of strict regular semigroups. For the intersection of WS with some usual e-varieties of regular semigroups, such as left regular orthodox semigroups and similar e-varieties V, we provide WS ⋂ V with a basis of identities and forbidden semigroups. The latter, by exclusion, characterize the given e-variety. We do this also for WS. Coupled with some results of Hall, Churchill and Trotter, we characterize in this way also the e-varieties of completely regular semigroups as well as regular semigroups which are locally completely regular. The e-varieties studied are depicted in a diagram.     

Regular F-Abundant Semigroups

ABSTRACT

The investigation of regular F-abundant semigroups is initiated. In fact, F-abundant semigroups are generalizations of regular cryptogroups in the class of abundant semigroups. After obtaining some properties of such semigroups, the construction theorem of the class of regular F-abundant semigroups is obtained. In addition, we also prove that a regular F-abundant semigroup is embeddable into a semidirect product of a regular band by a cancellative monoid. Our result is an analogue of that of Gomes and Gould on weakly ample semigroups, and also extends an earlier result of O'Carroll on F-inverse semigroups.     

正则矩阵半群

对于复数域上正则的矩阵半群S,证明如下各条是等价的: (1)S是(0-)单的;(2)S是(0-)单秩的; (3)S是完全(0-)单的.证明S的同态像中任意一个幂等元的下方必有本原幂等元;s的司态像若是(0-)单的则是完全(0-)单的.     

带正则*-断面的正则半群

本文给出了带正则*-断面的正则半群的若干性质,获得了带拟理想正则*-断面的正则半群的一个构造方法.利用这一构造定理,考虑了这类半群上的同余.     

Partial Orders on Transformation Semigroups

In 1986, Kowol and Mitsch studied properties of the so-called natural partial order on T(X), the total transformation semigroup defined on a set X. In particular, they determined when two total transformations are related under this order, and they described the minimal and maximal elements of (T(X), ). In this paper, we extend that work to the semigroup P(X) of all partial transformations of X, compare with another natural partial order on P(X), characterise the meet and join of these two orders, and determine the minimal and maximal elements of P(X) with respect to each order.This author gratefully acknowledges the generous support of Centro de Matematica, Universidade do Minho, Portugal during his visit in May–June 2001.Received May 27, 2002; in revised form November 27, 2002 Published online May 16, 2003     

Certain Partial Orders on Semigroups

Relations introduced by Conrad, Drazin, Hartwig, Mitsch and Nambooripad are discussed on general, regular, completely semisimple and completely regular semigroups. Special properties of these relations as well as possible coincidence of some of them are investigated in some detail. The properties considered are mainly those of being a partial order or compatibility with multiplication. Coincidences of some of these relations are studied mainly on regular and completely regular semigroups.     

Partial Orders on Transformation Semigroups

In 1986, Kowol and Mitsch studied properties of the so-called natural partial order on T(X), the total transformation semigroup defined on a set X. In particular, they determined when two total transformations are related under this order, and they described the minimal and maximal elements of (T(X), ). In this paper, we extend that work to the semigroup P(X) of all partial transformations of X, compare with another natural partial order on P(X), characterise the meet and join of these two orders, and determine the minimal and maximal elements of P(X) with respect to each order.     

关于Fuzzy正则半群

本文获得Ｆｕｚｚｙ正则半群中Ｆｕｚｚｙ理想的一些代数性质。     

S-Classification of Regular Semigroups

On any regular semigroup S, the greatest idempotent pure congruence τ the greatest idempotent separating congruence μ and the least band congruence β are used to give the S-classification of regular semigroups as follows. These congruences generate a sublattice Λ of the congruence lattice C(S) of S. We consider the triples (Λ,K,T), where K and T are the restrictions of the K- and T-relations on C(S) to Λ. Such triples are characterized abstractly and form the objects of a category S whose morphisms are surjective K- and T-preserving homomorphisms subject to a mild condition. The class of regular semigroups is made into a category S whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor from S to S. The effect of the S-classification on Reilly semigroups and cryptogroups is discussed briefly.     

T-Classification of Regular Semigroups

On any regular semigroup S, the least group congruence σ, the greatest idempotent separating congruence μ and the least band congruence β are used to give the T-classification of regular semigroups as follows. These congruences generate a sublattice Λ of the congruence lattice C(S) of S. We consider the triples (Λ,K,T), where K and T are the restrictions of the K- and T-relations on C(S) to Λ. Such triples are characterized abstractly and form the objects of a category T whose morphisms are surjective K-preserving homomorphisms subject to a mild condition. The class of regular semigroups is made into a category T whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor from T to T. The effect of the T-classification to P-semigroups is considered in some detail.     

E-酉正则半群

本文定义了PO-六元组的概念,给出了E-酉正则半群的一种结构.     

Flows on Regular Semigroups

We study the structure of the flow monoid of a regular semigroup. This arises from the approach of Nambooripad of considering a regular semigroup as a groupoid – a category in which every morphism is invertible. A flow is then a section to the source map in this groupoid, and the monoid structure of the set of all flows is determined in terms of the Green relations on the original semigroup.     

Eventually Regular Dubreil-Jacotin Semigroups

We derive necessary and sufficient conditions for an ordered eventually regular semigroup to be a Dubreil-Jacotin semigroup. We show that an eventually regular Dubreil-Jacotin semigroup has a biggest idempotent. Necessary and sufficient conditions for this biggest idempotent to be a weak middle unit or to be medial are related to the semigroup being naturally ordered and to the set of regular elements being a subsemigroup.     

E-special Ordered Regular Semigroups

An ordered regular semigroup S is E-special if for every x ∈ S there is a biggest x ⁺ ∈ S such that both xx ⁺ and x ⁺ x are idempotent. Every regular strong Dubreil–Jacotin semigroup is E-special, as is every ordered completely simple semigroup with biggest inverses. In an E-special ordered regular semigroup S in which the unary operation x → x ⁺ is antitone the subset P of perfect elements is a regular ideal, the biggest inverses in which form an inverse transversal of P if and only if S has a biggest idempotent. If S ⁺ is a subsemigroup and S does not have a biggest idempotent, then P contains a copy of the crown bootlace semigroup.     

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. May 24, 1999