Sequentially Injective Hull of Acts Over Idempotent Semigroups

In this paper using the notion of a sequentially dense monomorphism we consider sequential injectivity (s-injectivity) for acts over a semigroup S. Among other things we describe the s-injective hull of acts over an idempotent semigroup S. We also give some classes of idempotent semigroups for the acts over which the notions of injectivity and s-injectivity coincide, and hence get the injective hull of acts over these classes of semigroups.     

Characterizing Semigroups by Sequentially Dense Injective Acts

Sequentially dense monomorphisms and injectivity with respect to these monomorphisms were first introduced and studied by Giuli for acts over the monoid (N^∞, min). In this paper we generalize these notions to acts over a general semigroup, and study the behaviour of this notion of injectivity with respect to products, coproducts, and direct sums. As a result we give some characterizations of semigroups.     

Various notions of weak injectivity over Clifford semigroups

We study various notions of weak injectivity of acts over a Clifford semigroup. In addition we characterize weak self-injectivities of several classes of Clifford semigroups constructed via particular kinds of groups and group homomorphisms. Research of X. Zhang supported by the China Scholarship Council No. 2006101056. Research of Y. Wang supported by National Natural Science Foundation of China Research Grant 10571181.     

L－逆左系对幺半群的刻画

正则左Ｓ-系是ｖｏｎ Ｎｅｕｍａｎｎ正则半群的自然推广，逆左Ｓ-系是逆半群的自然推广．作为左逆半群的自然推广，本文引入了Ｌ-逆左系的概念，并用来刻画了几类幺半群，如左逆幺半群，逆幺半群，ａｄｅｑｕａｔｅ幺半群等．     

On the Baer Criterion for Acts Over Semigroups

It is known that although the Baer Criterion for injectivity holds for modules over rings with unit, it is not true for acts over an arbitrary monoid.

Seeking a characterization for the Baer Criterion to hold for acts over a monoid, in this article, using the notion of completeness introduced by Giuli, we find some classes of monoids such that for acts over them the Baer Criterion holds.     

The Booleanization of an inverse semigroup

Semigroup Forum - We prove that the forgetful functor from the category of Boolean inverse semigroups to the category of inverse semigroups with zero has a left adjoint. This left adjoint is what...     

Idempotent bounded C-semigroups

A semigroup with zero isidempotent bounded (IB) if it is the 0-direct union of idempotent generated principal left ideals and the 0-direct union of idempotent generated principal right ideals. Notable examples are completely 0-simple semigroups and the wider class of primitive abundant semigroups. Significant to the structure of these semigroups is that they are all categorical at zero. In this paper we describe IB semigroups that are categorical at zero in terms ofdouble blocked Rees matrix semigroups. This generalises Fountain's characterisation of primitive abundant semigroups via blocked Rees matrix semigroups [1], which in turn yields the Rees theorem for completely 0-simple semigroups.     

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l¹(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ?¹(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ?¹(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ?¹(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.     

On uniform acts over semigroups

The paper is devoted to the investigation of uniform acts over semigroups perceived as an overclass of subdirectly irreducible acts. We establish conditions for a uniform act to be subdirectly irreducible. In particular, we prove that uniform acts with two zeros are subdirectly irreducible. Ultimately we investigate monoids which are uniform as right acts over themselves and we describe regular monoids with this property.     

On the variational description of the trajectories of averaging quantum dynamical maps

We study the dynamics of quantum system with degenerated Hamiltonian. To this end we consider the approximating sequence of regularized Hamiltonians and corresponding sequence of dynamical semigroups acting in the space of quantum states. The limit points set of the sequence of regularized semigroups is obtained as the result of averaging by finitely additive measure on the set of regularizing parameters. We establish that the family of averaging dynamical maps does not possess the semigroup property and the injectivity property. We define the functionals on the space of maps of the time interval into the quantum states space such that the maximum points of this functionals coincide with the trajectories of the family of averaging dynamical maps.     

PERFECT rpp SEMIGROUPS

The aim of this paper is to study a class of rpp semigroups, namely the perfect rpp semigroups. We obtain some characterization theorems for such semigroups. In particular, the spined product structure of perfect rpp semigroups is established. As an application of spined product structure, we prove that a perfect rpp semigroup is a strong semilattice of left cancellative planks. By a left cancellative plank, we mean a product of a left cancellative monoid and a rectangular band. Thus, the work of J.B. Fountain on C-rpp semigroups is further developed.

     

The loop problem for Rees matrix semigroups

We study the relationship between the loop problem of a semigroup, and that of a Rees matrix construction (with or without zero) over the semigroup. This allows us to characterize exactly those completely zero-simple semigroups for which the loop problem is context-free. We also establish some results concerning loop problems for subsemigroups and Rees quotients.     

Solution of nonlinear stochastic equations with a generator of a semigroup discontinuous at zero

In this paper we study the Cauchy problem for a nonlinear stochastic equation with a generator of a semigroup discontinuous at zero. We prove the unique existence of a mild solution for some class of such semigroups. We establish a connection between weak and mild solutions for such semigroups.     

The ordinarization transform of a numerical semigroup and semigroups with a large number of intervals

A numerical semigroup is said to be ordinary if it has all its gaps in a row. Indeed, it contains zero and all integers from a given positive one. One can define a simple operation on a non-ordinary semigroup, which we call here the ordinarization transform, by removing its smallest non-zero non-gap (the multiplicity) and adding its largest gap (the Frobenius number). This gives another numerical semigroup and by repeating this transform several times we end up with an ordinary semigroup. The genus, that is, the number of gaps, is kept constant in all the transforms.This procedure allows the construction of a tree for each given genus containing all semigroups of that genus and rooted in the unique ordinary semigroup of that genus. We study here the regularity of these trees and the number of semigroups at each depth. For some depths it is proved that the number of semigroups increases with the genus and it is conjectured that this happens at all given depths. This may give some light to a former conjecture saying that the number of semigroups of a given genus increases with the genus.We finally give an identification between semigroups at a given depth in the ordinarization tree and semigroups with a given (large) number of gap intervals and we give an explicit characterization of those semigroups.     

左C-半群的左交错积结构

给出了左C-半群的另一种结构,所谓左交错积结构,并刻画了它的特殊情形.这种结构为左C-半群在广义正则半群类中的再推广奠定了基础.     

偏序半群的C-左理想

本文引入了偏序半群中Ｃ 左理想的概念，讨论了Ｃ 左理想的一些基本性质，定义了左基的概念并利用它给出了最大Ｃ 左理想存在的必要和充分条件．作为应用，本文还讨论了每个真左理想均为Ｃ 左理想和无Ｃ 左理想这两类半群的结构特征．本文的结果在一般半群中也成立     

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.