具有理想收缩性质的某些GV-半群(英文)

如果半群S的每一个理想都是它的幂等同态像,称半群S具有理想收缩性质。GV-半群是完全正则半群在π-正则半群范围内的推广。本文刻画了某些具有理想收缩性质的GV-半群。     

П正则半群的幂等元同余类

本文分别给出П正则半群的幂等元同余类和Пｏｒｔｈｏｄｏｘ半群［１］的幂等元同余类的П正则性刻画．其次，证明П逆半群或完全П正则半群Ｓ的幂等元同余类是Ｓ的П正则子半群．最后讨论ｏｒｔｈｏｄｏｘ半群的幂等元同合类的正则性．     

Products of idempotent matrices over integral domains

We say that a ring R has the idempotent matrices property if every square singular matrix over R is a product of idempotent matrices. It is known that every field, and more generally, every Euclidean domain has the idempotent matrices property. In this paper we show that not every integral domain has the idempotent matrices property and that if a projective free ring has the idempotent matrices property then it must be a Bezout domain. We also show that a principal ideal domain has the idempotent matrices property if and only if every fraction a/b with b≠0 has a finite continued fraction expansion. New proofs are also provided for the results that every field and every Euclidean domain have the idempotent matrices property.     

On the ranks of idempotent matrices over skew semifields

As is well known, every positive idempotent matrix is of rank 1. It is proved that idempotent matrices without zeros have this property over many skew semifields, and all these skew semifields are described.     

Reflexive Property of Rings

Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this article, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings.     

Almost Factorizable Orthodox Semigroups

In this paper, we investigate idempotent separating and arbitrary homomorphic images of semidirect products of bands by groups. We give characterizations for idempotent separating homomorphic images of semidirect products, and show that the class of all idempotent separating homomorphic images is strictly contained in the class of all homomorphic images. Furthermore, we give a characterization of all homomorphic images.     

A subdirectly irreducible band accepting a = a

Subdirectly irreducible bands which generate all proper subvarieties of bands (idempotent semigroups) were exhibited in [7]. In this paper a very general construction is presented which can be used to produce subdirectly irreducible bands which generate the variety of bands.     

Idempotent distributive semirings II

In [2] it is shown that every idempotent distributive semiring is the P?onka sum of a semilattice ordered system of idempotent distributive semirings which satisfy the generalized absorption law x+xyx+x=x. We shall show that an idempotent distributive semiring which satisfies the above absorption law must be a subdirect product of a distributive lattice and a semiring which satisfies the additional identity xyx+x+xyx=xyx. Using this, we construct the lattice of all equational classes of idempotent distributive semirings for which the two reducts are normal bands.     

Basic properties of e-inversive semigroups

Generalizing a property of regular resp. finite semigroups a semigroup S is called E-(0-) inversive if for every a ∈ S4(a ≠ 0) there exists x ∈ S such that ax (≠ 0) is an idempotent. Several characterizations are given allowing to identify the (completely, resp. eventually) regular semigroups in this class. The case that for every a ∈ S4(≠ 0) there exist x,y ∈ S such that ax = ya(≠ 0) is an idempotent, is dealt with also. Ideal extensions of E- (0-)inversive semigroups are studied discribing in particular retract extensions of completely simple semigroups. The structure of E- (0-)inversive semigroups satisfying different cancellativity conditions is elucidated. 1991 AMS classification number: 20M10.     

The Structure of Commutative Semigroups with the Ideal Retraction Property

A semigroup is said to have the ideal retraction property when each of its ideals is a homomorphic retraction of the whole semigroup. This paper presents a complete characterization of the commutative semigroups that enjoy this property. The fundamental building blocks of these semigroups are the 2-cores and the semilattice of idempotents. Structure for semilattices with the ideal retraction property was discussed in an earlier paper and the structure of the 2-core is described in detail within this paper.     

The Structure of NBe—rpp Semigroups

§ 1.ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭａｉｎＲｅｓｕｌｔｓ　ＡｓｅｍｉｇｒｏｕｐＳｉｓｃａｌｌｅｄａｎｒｐｐｓｅｍｉｇｒｏｕｐｉｆａｌｌｐｒｉｎｃｉｐａｌｒｉｇｈｔｉｄｅａｌｓａＳ1(ａ∈Ｓ)ｏｆＳ ,ｒｅｇａｒｄｅｄａｓａｎＳ1 ｓｙｓｔｅｍ ,ａｒｅｐｒｏｊｅｃｔｉｖｅ (ｓｅｅ,[1 ]ａｎｄ [2 ] ) .ＡｓｅｍｉｇｒｏｕｐＳｉｓａｎｒｐｐｓｅｍｉｇｒｏｕｐｉｆａｎｄｏｎｌｙｉｆｆｏｒａｌｌａ∈Ｓ ,ｔｈｅｓｅｔＭａ ={ｅ∈Ｅ Ｓ1ａ Ｓｅａｎｄ ( ｘ ,ｙ∈Ｓ1)ａｘ=ａｙ ｅｘ=ｅｙ}ｉｓｎｏｎｅｍｐｔｙ ,ｗｈｅｒｅＥ …     

On some classes of regular semigroups

An idempotent e of a semigroup S is called right [left] principal (B.R. Srinivasan, [2]) if fef=fe [fef=ef] for every idempotent f of S. Say that S has property (LR) [(LR₁)] if every ℒ-class of S contains atleast [exactly] one right principal idempotent. There and six further properties obtained by replacing, ‘ℒ-class’ by ‘ℛ-class’ and/or ‘right principal’ by ‘left principal’ are examined. If S has (LR₁), the set of right principal elementsa of S (aa′ is right principal for some inversea′ ofa) is an inverse subsemigroup of S, generalizing a theorem of Srinivasan [2] for weakly inverse semigroups. It is shown that the direct sum of all dual Schützenberger representations of an (LR) semigroup is faithful (cf[1], Theorem 3.21, p. 119). Finally, necessary and sufficient conditions are given on a regular subsemigroup S of the full transformation semigroup on a set in order that S has each of the properties (LR), (LR₁), etc.     

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.     

Note on the orderability of idempotent semigroups

In [3] , we gave a condition for the orderability of finite idempotent semigroups. Recently, in [5], T. Saitô gives a condition for the orderability of idempotent semigroups in the general case. As Corollary ( 4–13 of [5] ), he obtains an idempotent semigroup S is orderable if and only if every finite subsemigroup of S is orderable. The purpose of this note is to give a direct proof of this result.     

Semigroups with the ideal retraction property

In this paper we investigate the structure of semigroups with the ideal retraction property i.e., semigroups which are not simple and have the property that each ideal is a homomorphic retract of the semigroup. We present examples to show that the ideal retraction property is neither hereditary nor productive. That this property is preserved by homomorphisms is established for some classes of semigroups, but the general question remains open. The classes of semigroups investigated in this paper are separative semigroups, ideal semigroups, semilattices, cyclic semigroups, nil semigroups, and Clifford semigroups. It is established that a semigroup with zero 0 which is expressible as a direct sum of each ideal and a dual ideal (complement with 0 adjoined) has the ideal retraction property. The converse holds for ideal semigroups, and an example is presented which demonstrates that the converse does not hold in general.     

Commutative semigroups having greatest regular images

Let S be a commutative semigroup. S has a greatest regular image if and only if each of its Archimedean components contains an idempotent. S has a greatest group-with-zero image if and only if S has precisely two Archimedean components and the upper component contains an idempotent. The existence and structure of these images and of greatest group images is related to tensor products.     

Semigroups with the ideal retraction property

Abstract. In this paper we investigate the structure of semigroups with the ideal retraction property i.e., semigroups which are not simple and have the property that each ideal is a homomorphic retract of the semigroup. We present examples to show that the ideal retraction property is neither hereditary nor productive. That this property is preserved by homomorphisms is established for some classes of semigroups, but the general question remains open. The classes of semigroups investigated in this paper are separative semigroups, ideal semigroups, semilattices, cyclic semigroups, nil semigroups, and Clifford semigroups. It is established that a semigroup with zero 0 which is expressible as a direct sum of each ideal and a dual ideal (complement with 0 adjoined) has the ideal retraction property. The converse holds for ideal semigroups, and an example is presented which demonstrates that the converse does not hold in general.     

Modular varieties with the Fraser-Horn property

The notion of central idempotent elements in a ring can be easily generalized to the setting of any variety with the property that proper subalgebras are always nontrivial. We will prove that if such a variety is also congruence modular, then it has factorable congruences, i.e., it has the Fraser-Horn property. (This property is well known to have major implications for the structure theory of the algebras in the variety.)

     

An extremal property of the eigenvalue of irreducible matrices in idempotent algebra and solution of the Rawls location problem

An extremal property of the eigenvalue of an irreducible matrix in idempotent algebra is studied. It is shown that this value is the minimum value of some functional defined using this matrix on the set of vectors with nonzero components. The minimax problem of location of a single facility (the Rawls problem) on a plane with rectilinear distance is considered. For this problem, we give the corresponding representation in terms of idempotent algebra and suggest a new algebraic solution, which is based on the results of investigation of the extremal property of eigenvalue and reduces to finding the eigenvalue and eigenvectors of a certain matrix.     

Idempotent reducts of abelian groups

In this paper we construct two maps between the polynomials of abelian groups and the polynomials of idempotent reducts of abelian groups and show that these can be used to “lift” the finite basis property from abelian groups to their idenpotent reducts. It follows that this equational class of all idempotent reducts of abelian groups has the finite basis property. The research of this author was supported by a grant from the National Research Council of Canada. Presented by J. Mycielski