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.     

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。     

The Ideal Extension Property in Compact Semigroups

Abstract. The aim of this paper is to study and characterize compact semigroups with the ideal extension property. We establish a characterization of compact semigroups having the ideal extension property. In particular, we completely determine the structure of such semigroups with the property that regular elements form a subsemigroup, and also the structure of such semigroups with precisely one regular D-class.     

Globals of Idempotent Semigroups

If S is a semigroup, the global (or the power semigroup) of S is the set P(S) of all nonempty subsets of S equipped with a naturally defined multiplication. A class K of semigroups is globally determined if any two semigroups of K with isomorphic globals are themselves isomorphic. We study properties of globals of idempotent semigroups and show, in particular, that the class of normal bands is globally determined.     

Embedding Semigroups into Idempotent Generated Ones

We present a concrete model of the embedding due to Pastijn and Yan of a semigroup S into an idempotent generated semigroup now in terms of a Rees matrix semigroup over S¹. The paper starts with a comparison of the two embeddings. Studying the properties of this embedding, we prove that it is functorial. We show that a number of usual semigroup properties is preserved by this embedding, such as periodicity, finiteness, the cryptic property, regularity, complete semisimplicity and various local properties, but complete regularity is not one of them.     

Embedding Semigroups into Idempotent Generated Ones

We present a concrete model of the embedding due to Pastijn and Yan of a semigroup S into an idempotent generated semigroup now in terms of a Rees matrix semigroup over S¹. The paper starts with a comparison of the two embeddings. Studying the properties of this embedding, we prove that it is functorial. We show that a number of usual semigroup properties is preserved by this embedding, such as periodicity, finiteness, the cryptic property, regularity, complete semisimplicity and various local properties, but complete regularity is not one of them.     

严格部分变换半群的幂等元秩

设SP_n是X_n上的严格部分变换半群,对n≥4和2≤r≤n-2,证明了半群V(n,r)＝｛α∈SP_n∶｜imα｜≤r｝是幂等元生成,且秩和幂等元秩为(r+1)S(n,r+1).     

具有弱正规幂等元的富足半群的结构

本文研究含弱正规幂等元的富足半群.在给出这类半群的若干特征后,建立了具有弱正规幂等元的富足半群的结构．作为应用,给出具有正规幂等元的富足半群和具有（弱）正规幂等元的拟适当半群的结构．     

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.     

On Idempotent Ranks of Semigroups of Partial Transformations

A subset U of a semigroup S is a generating set for S if every element of S may be written as a finite product of elements of U. The rank of S is the size of a minimal generating set of S, and the idempotent rank of S is the size of a minimal generating set of S consisting of idempotents in S. A partition of a q-element subset of the set X_n={1,2,..., n} is said to be of type if the sizes of its classes form the partition of q n. A non-trivial partition of a positive integer q consists of k < q elements. For a non-trivial partition of q n, the semigroup S(), generated by all the transformations with kernels of type , is idempotent-generated. It is known that if is a non-trivial partition of n, that is, S() consists of total many-to-one transformations, then the rank and the idempotent rank of S() are both equal to max{ⁿ_d, N()}, where N() is the number of partitions of X_n of type . We extend this result to semigroups of partial transformations, and prove that if is a non-trivial partition of q < n, then the rank and the idempotent rank of S() are both equal to N().     

Some Semilattices of Semigroups Each Having One Idempotent

Research supported by the Italian Ministry of Universities and Scientific Research     

The Ideal Extension and Congruence Extension Properties for Compact Semigroups

The aim of this paper is to study the congruence extension property and the ideal extension property for compact semigroups. We present a characterization of compact semigroups with the ideal extension property and prove that each compact semigroup with the congruence extension property also has the ideal extension property.     

Ideal Extensions of Ordered Semigroups

Abstract

The ideal extensions of semigroups -without order- have been first considered by Clifford (Clifford, A. H. (1950). Extension of semigroups. Trans. Amer. Math. Soc. 68: 165–173). The aim of this paper is to give the main theorem of the ideal extensions for ordered semigroups.     

On Embeddability of Idempotent Separating Extensions of Inverse Semigroups

Let S be an inverse semigroup and rho an idempotent separating congruence on S. It is proved that S can be embedded into a lambda-semidirect product of a group F by S/rho where F belongs to the variety generated by the idempotent classes of rho.     

The Ideal Property and Traces

We prove a characterization of the ideal property for a large class of C*–algebras in terms of vanishing conditions for the finite traces on the algebra. An application of this result to a class of inductive limit C*–algebras (GAH algebras) is also given.     

The Spectral Mapping Property of Delay Semigroups

We offer a new way of proving spectral mapping properties of delay semigroups in L ^p -history spaces with finitely many rationally depending delays based on an explicit construction of approximate eigenvectors. This allows us to provide proper generalizations of the existing spectral mapping theorems. Submitted: April 25, 2007. Accepted: November 5, 2007.     

The Ideal Intersection Property for Groupoid Graded Rings

We show that, if a groupoid graded ring has a grading satisfying a certain nondegeneracy property, then the commutant of the center of the principal component of the ring has the ideal intersection property, that is it intersects nontrivially every nonzero ideal of the ring. Furthermore, we show that for skew groupoid algebras with commutative principal component, the principal component is maximal commutative if and only if it has the ideal intersection property.     

弱可约序半群的理想扩张

NioviKehayopulu和Michael Tsingelis于2003年给出了关于序半群的理想扩张的一个定理,本文利用该定理进一步给出了弱可约序半群的理想扩张的构造．     

A Combinatorial Property for Semigroups of Matrices

Abstract. In this paper we study the semigroups of matrices over a commutative semiring. We prove that a semigroup of matrices over a tropical semiring satisfies a combinatorial property called weak permutation property . We consider an application of this result to the Burnside problem for groups.     

Naturally Ordered Abundant Semigroups in Which Each Idempotent Has a Greatest Inverse

The aim of this article is to study naturally ordered abundant semigroups in which every idempotent has a greatest inverse. After obtaining some characterizations, their structure is established.