Pure ∇-Bands

Jonathan Leech proved in [8] that the multiplicative reduct of a skew lattice in a ring is always a regular band. A band that is closed under the operation ∇ is called a ∇-band. A ∇-band is also always regular. We show that a regular band does not generate a ∇-band in general, but does so in the special case of regular pure bands. An exact condition for a pure ∇-band to form a skew lattice was found in [1].     

正则带的拟强半格分解

本文研究了正则带的结构问题.利用拟强半格分解方法,获得了带为正则带当且仅当它为矩形带的拟强半格,推广了Yamada和Kimura关于正规带是矩形带的强半格的结果.     

On the structure of semigroups of idempotent matrices

We prove that any pure regular band of matrices admits a simultaneous LU decomposition in the standard form. In case that such a band forms a double band called a skew lattice, we obtain the standard form without the assumption of purity.     

Bands in lattices of operators

We consider the lattice of regular operators on a Dedekind complete Banach lattice. We show that in general the projection onto a band generated by a lattice homomorphism need not be continuous and that the principal bands need not be closed for the operator norm. In fact it is possible to find a convergent sequence of operators all the members of which are disjoint from the limit.

     

Exponential epigroups

We show that an exponential epigroup is a band of unipotent exponential epigroups (in which the set of idempotents forms a subsemigroup). Also, we investigate the structure of unipotent exponential epigroups. In addition, a characterization of the least normal band of groups congruence is given for an arbitrary medial epigroup. It is also shown that k-exponential eventually regular semigroups are necessarily epigroups.     

Varieties of idempotent semirings with commutative addition

The multiplicative reduct of an idempotent semiring with commutative addition is a regular band. Accordingly there are 13 distinct varieties consisting of idempotent semirings with commutative addition corresponding to the 13 subvarieties of the variety of regular bands. The lattice generated by the these 13 semiring varieties is described and models for the semirings free in these varieties are given. Received April 22, 2004; accepted in final form June 3, 2005.     

Ideal structure and factorization properties of the regular kernel operators

Positivity - We consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on $$L^p$$ -spaces. We show, in particular, that for any $$L^p(\mu )$$...     

Completely regular semigroups in the variety generated by the bicyclic semigroup

We shall show that a completely regular semigroup is in the semigroup variety generated by the bicyclic semigroup if and only if it is an orthogroup whose maximal subgroups are abelian. Therefore the lattice of subvarieties of the variety generated by the bicyclic semigroup contains as a sublattice a countably infinite distributive lattice of semigroup varieties, each of which consists of orthogroups with maximal subgroups that are torsion abelian groups. In particular, every band divides a power of the bicyclic semigroup.Presented by B. M. Schein.     

The lattice of idempotent distributive semiring varieties

A solution is given for the word problem for free idempotent distributive semirings. Using this solution the latticeL (ID) of subvarieties of the variety ID of idempotent distributive semirings is determined. It turns out thatL (ID) is isomorphic to the direct product of a four-element lattice and a lattice which is itself a subdirect product of four copies of the latticeL(B) of all band varieties. ThereforeL(ID) is countably infinite and distributive. Every subvariety of ID is finitely based. Project supported by the National Natural Science Foundation of China (Grant No. 19761004) and the Provincial Applied Fundamental Research Foundation of Yunnan (96a001z).     

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.     

左C-wrpp 半群的左$\Delta$-积

We study another structure of so-called left C-wrpp semigroups. In particular, the concept of left △-product is extended and enriched. The aim of this paper is to give a construction of left C-wrpp semigroups by a left regular band and a strong semilattice of left-R cancellative monoids. Properties of left C-wrpp semigroups endowed with left △-products are particularly investigated.     

A concept of variety for regular biordered sets

We define a bivariety of regular biordered sets to be a nonempty class of regular biordered sets which is closed under taking direct products, regular bimorphic images and relatively regular biordered subsets. It is then shown that there is a complete lattice morphism mapping the complete lattice of all e-varieties of regular semigroups onto the complete lattice of all bivarieties of regular biordered sets; as a corollary we prove that there is a complete lattice morphism mapping the complete lattice of all e-varieties of E-solid regular semigroups onto the complete lattice of all varieties of solid binary algebras. Examples of bivarieties include the class of all solid regular biordered sets and the class of all local semilattices. For each setX with at least two elements, we show that a bivariety contains a free object onX if and only if it consists entirely of solid regular biordered sets or it consists entirely of local semilattices. The author gratefully acknowledges the financial support of an Australian Postgraduate Research Award.     

Remarks on varieties of involution bands

In the present paper, we study varieties consisting of bands (idempotent semigroups) endowed with an involutorial antiautomorphism * as a fundamental operation. Our principal aim is here to provide an insight to some classes of these varieties from the structural point of view, especially in terms of semilattice decompositions, subdirect products and ideal extensions. In the course of such considerations, we shall extend the result of C. L. Adair [1], who described the lattice of all varieties of bands with a regular involution (i.e. with the identity x = xx ^* x) We depict a broader lattice of involution band varieties, which incorporates Adair’s lattice.     

幂格与商格的关系

该文在文［1］中幂格概念的基础上,得到了幂格的一个充要条件,给出了正则幂格、相对幂格的概念,并讨论了商格与正则幂格、相对幂格的关系.〖HT5”H〗关键词:〖HT5”SS〗格;幂格;商格;正则幂格;相对幂格.     

Split quasi-adequate semigroups

The so-called split IC quasi-adequate semigroups are in the class of idempotent-connected quasi-adequate semigroups. It is proved that an IC quasi-adequate semigroup is split if and only if it has an adequate transversal. The structure of such semigroup whose band of idempotents is regular will be particularly investigated. Our obtained results enrich those results given by McAlister and Blyth on split orthodox semigroups.     

The lattice of regular subsemigroups of a regular semigroup

It is shown that all regular subsemigroups of an arbitrary regular semigroup form a lattice. The properties of this lattice are investigated.