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.     

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 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.     

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     

On semigroups admitting ring structure

A multiplicative semigroup S with 0 is said to be a R-semigroup if S admits a ring structure. Isbell proved that if a finitely generated commutative semigroup is a R-semigroup, then it should be finite. The non-commutative version of this theorem is unsettled. This paper considers semigroups, not necessarily commutative, which are principally generated as a right ideal by single elements and semigroups which are generated by two independent generators and describes their structure. We also prove that if a cancellative 0-simple semigroup containing an identity is a R-semigroup, then it should be a group with zero. Communicated by A. H. Clifford     

On Ordered Semigroups Which are Semilattices of Simple and Regular Semigroups

We characterize the ordered semigroups which are decomposable into simple and regular components. We prove that each ordered semigroup which is both regular and intra-regular is decomposable into simple and regular semigroups, and the converse statement also holds. We also prove that an ordered semigroup S is both regular and intra-regular if and only if every bi-ideal of S is an intra-regular (resp. semisimple) subsemigroup of S. An ordered semigroup S is both regular and intra-regular if and only if the left (resp. right) ideals of S are right (resp. left) quasi-regular subsemigroups of S. We characterize the chains of simple and regular semigroups, and we prove that S is a complete semilattice of simple and regular semigroups if and only if S is a semilattice of simple and regular semigroups. While a semigroup which is both π-regular and intra-regular is a semilattice of simple and regular semigroups, this does not hold in ordered semigroups, in general.     

The Ideal Retraction Property for Idempotent Semigroups

A semigroup S is said to have the ideal retraction property provided each of its ideals is the image of an idempotent endomorphism of S. The main result of this work is a characterization of those bands which have the idempotent retraction property. All such bands are normal.     

Filters in (Quasiordered) Semigroups and Lattices of Filters

A filter in a semigroup is a subsemigroup whose complement is an ideal. (Alternatively, in a quasiordered semigroup, a slightly more general definition can be given.) We prove a number of results related to filters in a semigroup and the lattice of filters of a semigroup. For instance, we prove that every complete algebraic lattice can be the lattice of filters of a semigroup. We prove that every finite semigroup is a homomorphic image of a finite semigroup whose lattice of filters is boolean and which belongs to the pseudovariety generated by the original semigroup. We describe filter lattices of some well-known semigroups such as full transformation semigroups of finite sets (which are three-element chains) and free semigroups (which are boolean).     

左分式半群的几个性质

讨论了若干半群类在取左分式半群下的封闭性，给出了关于左发式半群的一个同构定理。     

Stabilizability of two classes of contraction semigroups

This paper studies the strong stabilizability of two classes of Hilbert space contraction semigroups: (i) strict contraction semigroups, which include those with strictly dissipative generators; and (ii) isometric or unitary semigroups. The former class is already weakly stable, while the latter is not strongly stable over the whole space. Our tool is the functional model of Hilbert space contractions; hence, strong stability of the semigroup is studied via stability of its cogenerator. It is shown that a strict contraction semigroup is, in general, not strongly stabilized by the feedback –B*, while an isometric or a unitary semigroup is strongly stabilized by the same feedback, providedB is not compact.     

Hungarian Semigrousps, the Arithmetric of Generalized Renewal Sequence and Semi-P-Functions

In this paper the authers discuss the additional conditions under which a Hungarian semigroup is possessed of the fundamental Delphic properties. It is proved that both the positive generalized renewal sequence semigroup and the tame semi-p-function semigroup are Hungarian semigroups and possessed of the fundamental Delphic properties. Then the arithmetic properties of these two classes of the special semigroups are studied respectively.     

平衡范畴与半群的双序

研究范畴与半群通过幂等元双序建立的一种自然联系.对每个有幂等元的半群S,其幂等元生成的左、右主理想之集通过双序ω~e,ω~r自然确定两个有子对象、有像且每个包含都右可裂的范畴L(S),R(S),其中态射的性质与S中元素的富足性、正则性有自然对应.利用这个联系,我们定义了"平衡(富足、正规)范畴"概念.对任一平衡(富足、正规)范畴■,我们构造其"锥半群"■,证明■左富足(富足、正则),且每个平衡(富足、正规)范畴■都与某左富足(富足、正则)半群S的左主理想范畴L(S)(作为有子对象的范畴)同构.     

Commutative semigroups whose homomorphic images in groups are groups

In this paper we study commutative semigroups whose every homomorphic image in a group is a group. We find that for a commutative semigroup S, this property is equivalent to S being a union of subsemigroups each of which either has a kernel or else is isomorphic to one of a sequence T₀, T₁, T₂, ... of explicitly given, countably infinite semigroups without idempotents. Moreover, if S is also finitely generated then this property is equivalent to S having a kernel.     

On the congruence extension property for compact semigroups

A topological semigroupS is said to have thecongruence extension property (CEP) provided that for each closed subsemigroupT ofS and each closed congruence σ onT, σ can be extended to a closed congruence onS. (That is, ∩(T xT=σ). The main result of this paper gives a characteriation of Γ-compact commutative archimedean semigroups with the congruence extension property (CEP). Consideration of this result was motivated by the problem of characterizing compact commutative semigroups with CEP as follows. It is well known that every commutative semigroup can be expressed as a semilattice of archimedean components each of which contains at most one idempotemt. The components of a compact commutative semigroup need not be compact (nor Γ-compact) as the congruence providing the decomposition is not necessarily closed. However, any component with CEP which is Γ-compact is characterized by the afore-mentioned result. Characterization of components of a compact commutative semigroup having CEP is a natural step towar characterization of the entire semigroup since CEP is a hereditary property. Other results prevented in this paper give a characterization of compact monothetic semigroups with CEP and show that Rees quotients of compact semigroups with CEP retain CEP.     

Orders in rings without identity

G. Lallement [5] proved that every idem potent congruence class of a regular semigroup contains an idem potent. P. Edwards [4] generalized this property of congruences to eventually regular semigroups. Using the natural partial order of the semigroup (see [6]) a weakened version of this result will be proved for the more general class of E-inversive semigroups. But for particular congruences the original result of Lallement still holds for every E-inversive semigroup. Finally, conditions for a congruence on a general semigroup (with E(S) a subsemigroup, resp.) are given, which ensure that Lallement's result holds.     

Tight inverse semigroups that are not bisimple

A semigroup is tight if each of its congruences is uniquely determined by each of the congruence classes. Bisimple inverse semigroups are tight, and tight semigroups are either simple or congruence-free with zero. Although congruence-free semigroups are tight, they are not necessarily bisimple. We construct tight inverse semigroups and tight inverse monoids that are neither bisimple nor congruence-free.     

Intuitionistic fuzzy semiprime ideals in ordered semigroups

We give characterizations of different classes of ordered semigroups by using intuitionistic fuzzy ideals. We prove that an ordered semigroup is regular if and only if every intuitionistic fuzzy left (respectively, right) ideal of S is idempotent. We also prove that an ordered semigroup S is intraregular if and only if every intuitionistic fuzzy two-sided ideal of S is idempotent. We give further characterizations of regular and intra-regular ordered semigroups in terms of intuitionistic fuzzy left (respectively, right) ideals. In conclusion of this paper we prove that an ordered semigroup S is left weakly regular if and only if every intuitionistic fuzzy left ideal of S is idempotent.     

The Lattice of Full Subsemigroups of an Inverse Semigroup

In this paper, we consider the lattice Subf S of full subsemigroups of an inverse semigroup S. Our first main theorem states that for any inverse semigroup S, Subf S is a subdirect product of the lattices of full subsemigroups of its principal factors, so that Subf S is distributive [meet semidistributive, join semidistributive, modular, semimodular] if and only if the lattice of full subsemigroups of each principal factor is. To examine such inverse semigroups, therefore, we need essentially only consider those which are 0-simple. For a 0-simple inverse semigroup S (not a group with zero), we show that in fact each of modularity, meet semidistributivity and join semidistributivity of Subf S is equivalent to distributivity of S, that is, S is the combinatorial Brandt semigroup with exactly two nonzero idempotents and two nonidempotents. About semimodularity, however, we concentrate only on the completely 0-simple case, that is, Brandt semigroups. For a Brandt semigroup S (not a group with zero), semimodularity of Subf S is equivalent to distributivity of Subf S. Finally, we characterize an inverse semigroup S for which Subf S is a chain.