scholarly journals The natural partial order on a regular semigroup

1980 ◽  
Vol 23 (3) ◽  
pp. 249-260 ◽  
Author(s):  
K. S. Subramonian Nambooripad
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Partial Order ◽  
Inverse Semigroups ◽  
Natural Partial Order ◽  
Simple Description ◽  
Regular Semigroups ◽  
Simple Congruence ◽  
Pseudo Inverse ◽  
Completely Simple

It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.

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.

A ∗-prehomomorphism of a type B semigroup

2020 ◽  
pp. 2150222
Author(s):  
Chunhua Li ◽  
Zhi Pei ◽  
Baogen Xu
Keyword(s):  
New York ◽  
Inverse Semigroup ◽  
Partial Order ◽  
Inverse Semigroups ◽  
Natural Partial Order ◽  
Type B ◽  
Abundant Semigroup ◽  
Basic Properties ◽  
Structure Theorems

Type B semigroups are generalizations of inverse semigroups, and every inverse semigroup admits an [Formula: see text]-unitary cover (M. Petrich, Inverse Semigroups (Wiley, New York, 1984)). Motivated by studying [Formula: see text]-unitary cover for inverse semigroups, and as a continuation of Petrich’s works in inverse semigroups, in this paper, we first introduce the concept of ∗-prehomomorphism of a type B semigroup. After obtaining some basic properties, we get some structure theorems and give some conditions for a type B semigroup which is constructed by using the ∗-prehomomorphism to be proper. In particular, we introduce the notion of [Formula: see text]-unitary good cover for an abundant semigroup, and prove that every type B semigroup with compatible natural partial order admits an [Formula: see text]-unitary good cover.

scholarly journals Epis are onto for finite regular semigroups

1983 ◽  
Vol 26 (2) ◽  
pp. 151-162 ◽  
Author(s):  
T. E. Hall ◽  
P. R. Jones
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Open Problem ◽  
Inverse Semigroups ◽  
Regular Semigroups ◽  
Preliminary Results ◽  
Absolutely Closed ◽  
Finite Regular Semigroup

After preliminary results and definitions in Section 1, we show in Section 2 that any finite regular semigroup is saturated, in the sense of Howie and Isbell [8] (that is, the dominion of a finite regular semigroup U in a strictly containing semigroup S is never S). This is equivalent of course to showing that in the category of semigroups any epi from a finite regular semigroup is in fact onto. Note for inverse semigroups the stronger result, that any inverse semigroup is absolutely closed [11, Theorem VII. 2.14] or [8, Theorem 2.3]. Further, any inverse semigroup is in fact an amalgamation base in the class of semigroups [10], in the sense of [5]. These stronger results are known to be false for finite regular semigroups [8, Theorem 2.9] and [5, Theorem 25]. Whether or not every regular semigroup is saturated is an open problem.

scholarly journals JOINS AND COVERS IN INVERSE SEMIGROUPS AND TIGHT -ALGEBRAS

2014 ◽  
Vol 90 (1) ◽  
pp. 121-133 ◽  
Author(s):  
ALLAN P. DONSIG ◽  
DAVID MILAN
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Inverse Semigroups ◽  
Natural Partial Order ◽  
Higher Rank ◽  
Special Case

AbstractWe show Exel’s tight representation of an inverse semigroup can be described in terms of joins and covers in the natural partial order. Using this, we show that the ${C}^{\ast } $-algebra of a finitely aligned category of paths, developed by Spielberg, is the tight ${C}^{\ast } $-algebra of a natural inverse semigroup. This includes as a special case finitely aligned higher-rank graphs: that is, for such a higher-rank graph $\Lambda $, the tight ${C}^{\ast } $-algebra of the inverse semigroup associated to $\Lambda $ is the same as the ${C}^{\ast } $-algebra of $\Lambda $.

Embedding Regular Semigroups into Idempotent Generated Ones

2010 ◽  
Vol 17 (02) ◽  
pp. 229-240 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Natural Partial Order ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Close Relationship ◽  
Completely Regular Semigroups ◽  
Relationship Of ◽  
And Inversion ◽  
The Relationship

Any semigroup S can be embedded into a semigroup, denoted by ΨS, having some remarkable properties. For general semigroups there is a close relationship between local submonoids of S and of ΨS. For a number of usual semigroup properties [Formula: see text], we prove that S and ΨS simultaneously satisfy [Formula: see text] or not. For a regular semigroup S, the relationship of S and ΨS is even closer, especially regarding the natural partial order and Green's relations; in addition, every element of ΨS is a product of at most four idempotents. For completely regular semigroups S, the relationship of S and ΨS is still closer. On the lattice [Formula: see text] of varieties of completely regular semigroups [Formula: see text] regarded as algebras with multiplication and inversion, by means of ΨS, we define an operator, denoted by Ψ. We compare Ψ with some of the standard operators on [Formula: see text] and evaluate it on a small sublattice of [Formula: see text].

scholarly journals Inverse semigroups determined by their partial automorphism monoids

2006 ◽  
Vol 81 (2) ◽  
pp. 185-198 ◽  
Author(s):  
Simon M. Goberstein
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Inverse Semigroups ◽  
Natural Partial Order ◽  
Inverse Monoid ◽  
Inverse Monoids ◽  
Inverse Subsemigroups

AbstractThe partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup S with no isolated nontrivial subgroups is lattice determined ‘modulo semilattices’ and if T is an inverse semigroup whose partial automorphism monoid is isomorphic to that of S, then either S and T are isomorphic or they are dually isomorphic chains relative to the natural partial order; a similar result holds if T is any semigroup and the inverse monoids consisting of all isomorphisms between subsemigroups of S and T, respectively, are isomorphic. Moreover, for these results to hold, the conditions that S be tightly connected and have no isolated nontrivial subgroups are essential.

scholarly journals Coextensions of pseudo-Inverse semigroups by rectangular bands

1980 ◽  
Vol 30 (1) ◽  
pp. 73-86 ◽  
Author(s):  
John Meakin
Keyword(s):  
Regular Semigroup ◽  
Rectangular Band ◽  
Inverse Semigroups ◽  
Regular Semigroups ◽  
Pseudo Inverse

AbstractWe say that a regulär semigroup S is a coetension of a (regular) semigroup T by rectangular bands if there is a homomorphism ϕ: S → T from S onto T such that, for each e = e2 ∈ S, e(ϕ ϕ-1) is a rectangular band. Regular semigroups which are coextesions of pseudo-inverse semigroups by rectangular bands may be characterized as those regular semigroups S with the property that, for each e = e2 ∈ S, ω(e) = {f = f2 ∈ S: ef = f} and ωl(e) = {f = f2 ∈ S: fe = f} are bands: this paper is concerned with a study of such semigroups.

scholarly journals Maximum idempotents in naturally ordered regular semigroups

1983 ◽  
Vol 26 (2) ◽  
pp. 213-220 ◽  
Author(s):  
D. B. McAlister ◽  
R. McFadden
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Natural Partial Order ◽  
Ordered Semigroup ◽  
Regular Semigroups ◽  
Partially Ordered ◽  
Image Position ◽  
Reverse Implication

We shall denote by ω the natural partial order on the idempotents E = E(S) of a regular semigroup S, so that in E,A partially ordered semigroup S(≦) is called naturally partially ordered [9] if the imposed partial order ≦ extends ω in the sense thatNo assumption is made about the reverse implication.

scholarly journals Inverse semigroups with isomorphic partial automorphism semigroups

1989 ◽  
Vol 47 (3) ◽  
pp. 399-417 ◽  
Author(s):  
Simon M. Goberstein
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Modular Lattice ◽  
Inverse Semigroups ◽  
Natural Partial Order ◽  
Dual Isomorphism ◽  
Combinatorial Inverse Semigroup ◽  
Inverse Subsemigroups

AbstractIt is shown that a so-called shortly connected combinatorial inverse semigroup is strongly lattice-determined “modulo semilattices”. One of the consequences of this theorem is the known fact that a simple inverse semigroup with modular lattice of full inverse subsemigroups is strongly lattice-determined [7]. The partial automorphism semigroup of an inverse semigroup S consists of all isomorphisms between inverse subsemigroups of S. It is proved that if S is a shortly connected combinatorial inverse semigroup, T an inverse semigroup and the partial automorphism semigroups of S and T are isomorphic, then either S and T are isomorphic or they are dually isomorphic chains (with respect to the natural partial order); moreover, any isomorphism between the partial automorphism semigroups of S and T is induced either by an isomorphism or, if S and T are dually isomorphic chains, by a dual isomorphism between S and T. Counter-examples are constructed to demonstrate that the assumptions about S being shortly connected and combinatorial are essential.

Rees matrix semigroups over inverse semigroups

1986 ◽  
Vol 102 (1-2) ◽  
pp. 61-90 ◽  
Author(s):  
F. J. Pastijn ◽  
Mario Petrich
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Rees Matrix Semigroup ◽  
Regular Semigroups ◽  
Rees Matrix Semigroups ◽  
Special Cases ◽  
Matrix Semigroup ◽  
Matrix Semigroups ◽  
Completely Simple

SynopsisA Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

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