scholarly journals On 𝓠-regular semigroups

2018 ◽  
Vol 16 (1) ◽  
pp. 522-530
Author(s):  
Xinyang Feng
Keyword(s):  
Direct Product ◽  
Homomorphic Image ◽  
Regular Semigroups ◽  
Subdirect Products ◽  
Maximum Group

AbstractIn this paper, we give some characterizations of 𝓠-regular semigroups and show that the class of 𝓠-regular semigroups is closed under the direct product and homomorphic images. Furthermore, we characterize the 𝓠-subdirect products of this class of semigroups and study the E-unitary 𝓠-regular covers for 𝓠-regular semigroups, in particular for those whose maximum group homomorphic image is a given group. As an application of these results, we claim that the similar results on V-regular semigroups also hold.

scholarly journals Subdirect products of E–inversive semigroups

1990 ◽  
Vol 48 (1) ◽  
pp. 66-78 ◽  
Author(s):  
H. Mitsch
Keyword(s):  
Explicit Form ◽  
Direct Product ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Group Congruence ◽  
Subdirect Products ◽  
Special Case ◽  
Maximum Group

AbstractA semigroup S is called E-inversive if for every a ∈ S ther is an x ∈ S such that (ax)2 = ax. A construction of all E-inversive subdirect products of two E-inversive semigroups is given using the concept of subhomomorphism introduced by McAlister and Reilly for inverse semigroups. As an application, E-unitary covers for an E-inversive semigroup are found, in particular for those whose maximum group homomorphic image is a given group. For this purpose, the explicit form of the least group congruence on an arbitrary E-inversive semigroup is given. The special case of full subdirect products of a semilattice and a group (that is, containing all indempotents of the direct product) is investigated and, following an idea of Petrich, a construction of all these semigroups is provided. Finally, all periodic semigroups which are subdirect products of a semilattice or a band with a group are characterized.

scholarly journals Regular semigroups, fundamental semigroups and groups

1980 ◽  
Vol 29 (4) ◽  
pp. 475-503 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Direct Product ◽  
Homomorphic Image ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Natural Partial Order ◽  
Regular Semigroups ◽  
Necessary And Sufficient ◽  
Fundamental Semigroups

AbstractIn this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ: S → T between regular semigroups. This is a mapping such that (ab) θ ≦ aθ bθ in the natural partial order on T.

scholarly journals On the lattice of varieties of completely regular semigroups

1983 ◽  
Vol 35 (2) ◽  
pp. 227-235 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Lattice Of Varieties ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Variety Of Groups ◽  
Completely Regular Semigroups ◽  
Subdirect Products

AbstractSeveral morphisms of this lattice V(CR) are found, leading to decompostions of it, and various sublattices, into subdirect products of interval sublattices. For example the map V → V ∪ G (where G is the variety of groups) is shown to be a retraction of V(CR); from modularity of the lattice V(BG) of varieties of bands of groups it follows that the map V → (V ∪ V V G) is an isomorphism of V(BG).

Additive Maps on Units of Rings

2018 ◽  
Vol 61 (1) ◽  
pp. 130-141
Author(s):  
Tamer Košan ◽  
Serap Sahinkaya ◽  
Yiqiang Zhou
Keyword(s):  
Recent Result ◽  
Direct Product ◽  
Homomorphic Image ◽  
Semilocal Ring ◽  
Additive Map ◽  
Exchange Rings ◽  
Additive Maps

AbstractLet R be a ring. A map f: R → R is additive if f(a + b) = f(a) + f(b) for all elements a and b of R. Here, a map f: R → R is called unit-additive if f(u + v) = f(u) + f(v) for all units u and v of R. Motivated by a recent result of Xu, Pei and Yi showing that, for any field F, every unit-additive map of (F) is additive for all n ≥ z, this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring R is additive if and only if either R has no homomorphic image isomorphic to or R/J(R) ≅ with 2 = 0 in R. Consequently, for any semilocal ring R, every unit-additive map of (R) is additive for all n ≥ 2. These results are further extended to rings R such that R/J(R) is a direct product of exchange rings with primitive factors Artinian. A unit-additive map f of a ring R is called unithomomorphic if f(uv) = f(u)f(v) for all units u, v of R. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

Inverse semigroups generated by a pair of subgroups

1977 ◽  
Vol 77 (1-2) ◽  
pp. 9-22 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Homomorphic Image ◽  
Main Part ◽  
Structure Theorem ◽  
Inverse Semigroups ◽  
Infinite Cyclic Group ◽  
The Third ◽  
Group A ◽  
Maximum Group

SynopsisThe aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.

Structure of Strict Abundant Semigroups

2019 ◽  
Vol 26 (03) ◽  
pp. 387-400
Author(s):  
Yizhi Chen ◽  
Bo Yang ◽  
Aiping Gan
Keyword(s):  
Tree Structure ◽  
General Construction ◽  
Regular Semigroups ◽  
New Class ◽  
Abundant Semigroups ◽  
Structure Theorems ◽  
Subdirect Products

We introduce a new class of semigroups called strict abundant semigroups, which are concordant semigroups and subdirect products of completely [Formula: see text]-simple abundant semigroups and completely 0-[Formula: see text]-simple primitive abundant semigroups. A general construction and a tree structure of such semigroups are established. Consequently, the corresponding structure theorems for strict regular semigroups given by Auinger in 1992 and by Grillet in 1995 are generalized and extended. Finally, an example of strict abundant semigroups is also given.

scholarly journals Product structure in topological algebras

1975 ◽  
Vol 20 (4) ◽  
pp. 385-393
Author(s):  
Desmond A. Robbie
Keyword(s):  
Direct Product ◽  
Homomorphic Image ◽  
Product Structure ◽  
Topological Algebras ◽  
Free Semigroups

It is shown that every compact nonconnected semigroup (semiring) which has commuting congruences, has a nontrivial continous homomorphic image which is iseomorphic to a direct product of finite congruence free semigroups (semirings). (This extends parts of earlier work by Kaplansky (1947) on compact rings.) It is also shown that there is a possibly finer representation but onto a product of congruence free semigroups (semirings) known only to be compact Hausdorff. A number of the techniques used evolve from work of Professor Wallace, who retired in mid-1973, and to whom this paper is dedicated.

Regular semigroups which are subdirect products of a band and a semilattice of groups

1973 ◽  
Vol 14 (1) ◽  
pp. 27-49 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Special Cases ◽  
Completely Regular Semigroups ◽  
Semilattice Of Semigroups ◽  
Subdirect Products ◽  
Completely Simple

In the study of the structure of regular semigroups, it is customary to impose several conditions restricting the behaviour of ideals, idempotents or elements. In a few instances, one may represent them as subdirect products of some much more restricted types of regular semigroups, e.g., completely (0-) simple semigroups, bands, semilattices, etc. In particular, studying the structure of completely regular semigroups, one quickly distinguishes certain special cases of interest when these semigroups are represented as semilattices of completely simple semigroups. In fact, this semilattice of semigroups may be built in a particular way, idempotents may form a subsemigroup, ℋ may be a congruence, and so on.

scholarly journals Congruence lattices of regular semigroups related to kernels and traces

1991 ◽  
Vol 34 (2) ◽  
pp. 179-203 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Distributive Lattice ◽  
Regular Semigroup ◽  
Homomorphic Image ◽  
Regular Semigroups ◽  
Generators And Relations ◽  
Congruence Lattices ◽  
Left And Right ◽  
Lattice Of Congruences

The kernel–trace approach to congruences on a regular semigroup S can be refined by introducing the left and right traces. This induces eight operators on the lattice of congruences on S: t1, k, tr,; Tt, K, Tr; t, T. We describe the lattice of congruences on S generated by six 3-element subsets of the set {ωt1, ωk, ωtr, εTt, εK, εTr} where ω and ε denote the universal and the equality relations. This is effected by means of a diagram and in terms of generators and relations on a free distributive lattice, or a homomorphic image thereof. We perform the same analysis for the lattice of congruences on S generated by the set {εK, ωk, εT, ωt}.

scholarly journals The modularity of the lattice of varieties of completely regular semigroups and related representations

1990 ◽  
Vol 32 (2) ◽  
pp. 137-152 ◽  
Author(s):  
Mario Petrich ◽  
Norman R. Reilly
Keyword(s):  
Regular Semigroup ◽  
Unary Operation ◽  
Direct Products ◽  
Completely Regular Semigroup ◽  
Lattice Of Varieties ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups ◽  
Subdirect Products ◽  
Image Position

A semigroup endowed with a unary operation satisfying the identitiesis a completely regular semigroup. In several recent papers devoted to the study of the lattice of subvarieties of the variety of completely regular semigroups, various results have been obtained which decompose special intervals in into either direct products or subdirect products. Petrich [14], Hall and Jones [6] and Rasin [20] have shown that certain intervals of the form , where is the trivial variety and are subdirect products of and Pastijn and Trotter [13] show that certain intervals of the form are direct products of the intervals and The main objective of this paper is to develop an appropriate lattice theoretic framework for these representations.

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