On a Problem of M. Kambites Regarding Abundant Semigroups

A semigroup is regular if it contains at least one idempotent in each ?-class and in each ?-class. A regular semigroup is inverse if it satisfies either of the following equivalent conditions: (i) there is a unique idempotent in each ?-class and in each ?-class, or (ii) the idempotents commute. Analogously, a semigroup is abundant if it contains at least one idempotent in each ?*-class and in each ?*-class. An abundant semigroup is adequate if its idempotents commute. In adequate semigroups, there is a unique idempotent in each ?* and ?*-class. M. Kambites raised the question of the converse: in a finite abundant semigroup such that there is a unique idempotent in each ?* and ?*-class, must the idempotents commute? In this note, we provide a negative answer to this question.     

On the Embedding of Ordered Semigroups into Ordered Group

It was shown in [7] that any right reversible, cancellative ordered semigroup can be embedded into an ordered group and as a consequence, it was shown that a commutative ordered semigroup can be embedded into an ordered group if and only if it is cancellative. In this paper we introduce the concept of L-maher and R-maher semigroups and use a technique similar to that used in [7] to show that any left reversible cancellative ordered L or R-maher semigroup can be embedded into an ordered group.     

Regularity on Near Permutation Semigroups

Abstract

A transformation semigroup over a set X with N elements is said to be a near permutation semigroup if it is generated by a group of permutations on N elements and by a set of transformations of rank N ? 1. The aim of this paper is to determine computationally efficient conditions to test whether or not a near permutation semigroup is regular.     

The structure of superabundant semigroups

A structure theorem for superabundant semigroups in terms of semilattices of normalized Rees matrix semigroups over some cancellative monoids is obtained. This result not only provides a construction method for superabundant semigroups but also generalizes the well-known result of Petrich on completely regular semigroups. Some results obtained by Fountain on abundant semigroups are also extended and strengthened.     

Medial permutable semigroups of the first kind

We say that a semigroup S is a permutable semigroup if the congruences of S commute with each other, that is, α○β=β○α is satisfied for all congruences α and β of S. A semigroup is called a medial semigroup if it satisfies the identity axyb=ayxb. The medial permutable semigroups were examined in Proc. Coll. Math. Soc. János Bolyai, vol. 39, pp. 21–39 (1981), where the medial semigroups of the first, the second and the third kind were characterized, respectively. In Atta Accad. Sci. Torino, I-Cl. Sci. Fis. Mat. Nat. 117, 355–368 (1983) a construction was given for medial permutable semigroups of the second [the third] kind. In the present paper we give a construction for medial permutable semigroups of the first kind. We prove that they can be obtained from non-archimedean commutative permutable semigroups (which were characterized in Semigroup Forum 10, 55–66, 1975). Research supported by the Hungarian NFSR grant No T042481 and No T043034.     

Tensor products of recurrent hypercyclic semigroups

We study tensor products of strongly continuous semigroups on Banach spaces that satisfy the hypercyclicity criterion, the recurrent hypercyclicity criterion or are chaotic.     

On Super Hamiltonian Semigroups

The concept of super hamiltonian semigroup is introduced. As a result, the structure theorems obtained by A. Cherubini and A. Varisco on quasi commutative semi-groups and quasi hamiltonian semigroups respectively are extended to super hamiltonian semigroups.     

Regularity and Unit-regularity of Generalized Semigroups of Linear Transformations

Let V and W be vector spaces over a division ring D and L_D (V, W) the set of all linear transformations from V into W. For L_D(W, V), let (L_D (V, W), ) denote the semigroup L_D (V, W) with the operation * defined by * = for all , L_D(V, W). By a unit-regular semigroup we mean a semigroup S with identity having the property that for each a S, a = aua for some unit u S. The main purpose of this paper is to prove the following statements. The semigroup (L_D(V, W), ) is regular if and only if V = {0}, W = {0} or is an isomorphism from W onto V. The semigroup (L_D (V, W), ) is unit-regular if and only if (i) V = {0}, (ii) W = {0} or (iii) is an isomorphism from W onto V and dim_D V .     

Left Abundant Semigroups

Abstract

The aim of this paper is to study some special lpp-semigroups, namely, the left GC-lpp semigroups. After obtaining some properties and characterizations of such semigroups, we establish some structure theorems of this class of semigroups. In addition, we also consider some special cases. As an application, we describe the structure theorems of IC quasi-adequate semigroups whose idempotent band is a regular band.     

基半群为0-J-严格单半群的零直并的代数

设Ａ是代数闭域ｋ上的一个具乘基Ｂ的有限维含幺结合代数，称半群Ｂ∪｛０｝为Ａ的基半群．本文给出了０ Ｊ 严格单半群的定义．对于基半群为０ Ｊ 严格单半群的零直并的代数，完全研究了它的代数表示型     

Primary Ideals and Valuation Ideals in Semigroups

Let S be a cancellation torsion-free additive semigroup with identity 0 and let S {0}. We consider the relation between the valuation ideals and primary ideals of S. We give a characterization that each primary ideal is a valuation ideal in S and also give a characterization that each valuation ideal is a primary ideal in S.AMS Subject Classification (1991): 20M14     

紧致交换矩阵半群

通过将矩阵同时对角化或同时上三角化的方法,给出有关紧致Abel矩阵半群以及紧致Hermite矩阵半群中矩阵的特征值的一些很好的刻画,证明了由可逆的Hermite矩阵构成的紧致矩阵半群中每个矩阵的特征值都是±1,Hermite矩阵单半群相似于对角矩阵半群,紧致交换矩阵半群的谱半径不超过1,等等.     

Global convoluted semigroups

Global exponentially bounded convoluted semigroups in Banach spaces are systematically treated with the help of Laplace transform. A perturbation theorem in this context is proved and some characterizations of the introduced class of analytic convoluted semigroups are obtained. Illustrative examples of generators of convoluted semigroups, including differential operators, are presented. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Classes of E-Inversive Semigroups and Semigroups Whose Idempotents Form a Subsemigroup

Abstract

A semigroup S is called E-inversive if for every a ∈ S there is an x ∈ S such that ax is idempotent. The purpose of this paper is the investigation of E-inversive semigroups and semigroups whose idempotents form a subsemigroup. Basic properties are analysed and, in particular, semigroups whose idempotents form a semilattice or a rectangular band are considered. To provide examples and characterizations, the construction methods of generalized Rees matrix semigroups and semidirect products are employed.     

A note on the power semigroup of a completely simple semigroup

We prove the pseudovariety generated by power semigroups of completely simple semigroups is the semidirect product of the pseudovariety of block groups with the pseudovariety of right zero semigroups, and hence is decidable. This answers a question of Almeida from over 15 years ago. The author was supported in part by NSERC.     

Tight representations of semilattices and inverse semigroups

By a Boolean inverse semigroup we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse semigroup in a Boolean inverse semigroup which are tight in a certain well defined technical sense. These representations are supposed to preserve as much as possible any trace of Booleanness present in the semilattice of idempotents of  . After observing that the Vagner–Preston representation is not tight, we exhibit a canonical tight representation for any inverse semigroup with zero, called the regular tight representation. We then tackle the question as to whether this representation is faithful, but it turns out that the answer is often negative. The lack of faithfulness is however completely understood as long as we restrict to continuous inverse semigroups, a class generalizing the E ^*-unitaries. Partially supported by CNPq.     

纯整Ehresmann半群的若干刻划

本文介绍纯整Ehresmann半群．纯整Ehresmann半群是一类特殊的U-半富足半群，我们给出了这类半群的若干刻划，并讨论了一些特殊的纯整Ehresmann半群．     

Supercyclic and chaotic translation semigroups

We give a necessary and sufficient condition for a translation semigroup to be supercyclic and to be chaotic in a weighted function space.