scholarly journals On quasi-orthodox semigroups with inverse transversals

1997 ◽  
Vol 40 (3) ◽  
pp. 505-514 ◽  
Author(s):  
T. S. Blyth ◽  
M. H. Almeida Santos
Keyword(s):  
Regular Semigroup ◽  
Inverse Transversal ◽  
Inverse Subsemigroup ◽  
Orthodox Semigroups

An inverse transversal of a regular semigroup S is an inverse subsemigroup So that contains precisely one inverse of each element of S. Here we consider the case where S is quasi-orthodox. We give natural characterisations of such semigroups and consider various properties of congruences.

scholarly journals Congruences on orthodox semigroups with associate subgroups

1996 ◽  
Vol 38 (1) ◽  
pp. 113-124
Author(s):  
T. S. Blyth ◽  
Emília Giraldes ◽  
M. Paula O. Marques-Smith
Keyword(s):  
Regular Semigroup ◽  
Maximal Subgroup ◽  
Structure Theorem ◽  
Inverse Transversal ◽  
Similar Concept ◽  
Inverse Subsemigroup ◽  
Associate Subgroup ◽  
Orthodox Semigroups ◽  
Middle Unit

If Sis a regular semigroup then an inverse transversal of S is an inverse subsemigroup T with the property that |T ∩ V(x)| = 1 for every x ∈ S where V(x) denotes the set of inverses of x ∈ S. In a previous publication [1] we considered the similar concept of a subsemigroup T of S such that |T ∩ A(x)| = 1 for every x ∩ S where A(x) = {y∈ S;xyx = x} denotes the set of associates (or pre-inverses) of x ∈ S, and showed that such a subsemigroup T is necessarily a maximal subgroup Ha for some idempotent α ∈ S. Throughout what follows, we shall assume that S is orthodox and α is a middle unit (in the sense that xαy = xy for all x, y ∈ S). Under these assumptions, we obtained in [1] a structure theorem which generalises that given in [3] for uniquely unit orthodox semigroups. Adopting the notation of [1], we let T ∩ A(x) = {x*} and write the subgroup T as Hα = {x*;x ∈ S}, which we call an associate subgroup of S. For every x ∈ S we therefore have x*α = x* = αx* and x*x** = α = x**x*. As shown in [1, Theorems 4, 5] we also have (xy)* = y*x* for all x, y ∈ S, and e* = α for every idempotent e.

scholarly journals A simplistic approach to inverse transversals

1996 ◽  
Vol 39 (1) ◽  
pp. 57-69 ◽  
Author(s):  
T. S. Blyth ◽  
M. H. Almeida Santos
Keyword(s):  
Regular Semigroup ◽  
Inverse Transversal ◽  
Inverse Subsemigroup

An inverse transversal of a regular semigroup S is an inverse subsemigroup that contains precisely one inverse of each element of S. In the literature there are three known types of inverse transversal, namely those that are multiplicative, those that are weakly multiplicative, and those that form quasi-ideals. Here, by considering natural ways in which certain words can be simplified, we reveal four new types of inverse transversal. All of these can be illustrated nicely in examples that are based on 2 × 2 matrices.

Regular semigroups with a multiplicative inverse transversal

1982 ◽  
Vol 92 (3-4) ◽  
pp. 253-270 ◽  
Author(s):  
T. S. Blyth ◽  
R. McFadden
Keyword(s):  
Regular Semigroup ◽  
Inverse Transversal ◽  
Multiplicative Inverse ◽  
Multiplicative Property ◽  
Regular Semigroups ◽  
Inverse Subsemigroup

SynopsisBy an inverse transversal of a regular semigroup S we mean an inverse subsemigroup that contains a single inverse of every element of S. A certain multiplicative property (which in the case of a band is equivalent to normality) is imposed on an inverse transversal and a complete description of the structure of S is obtained.

scholarly journals Construction of regular semigroups with inverse transversals

1989 ◽  
Vol 32 (1) ◽  
pp. 41-51 ◽  
Author(s):  
Tatsuhiko Saito
Keyword(s):  
Regular Semigroup ◽  
Unique Element ◽  
Inverse Transversal ◽  
Regular Semigroups ◽  
Inverse Subsemigroup

Let S be a regular semigroup. An inverse subsemigroup S° of S is an inverse transversal if |V(x)∩S°| = 1 for each x∈S, where V(x) denotes the set of inverses of x. In this case, the unique element of V(x)∩S° is denoted by x°, and x°° denotes (x°)–1. Throughout this paper S denotes a regular semigroup with an inverse transversal S°, and E(S°) = E° denotes the semilattice of idempotents of S°. The sets {e∈S:ee° = e} and {f∈S:f°f=f} are denoted by Is and Λs, respectively, or simply I and Λ. Though each element of these sets is idempotent, they are not necessarily sub-bands of S. When both I and Λ are sub-bands of S, S° is called an S-inverse transversal. An inverse transversal S° is multiplicative if x°xyy°∈E°, and S° is weakly multiplicative if (x°xyy°)°∈E° for every x, y∈S. A band B is left [resp. right] regular if e f e = e f [resp. e f e = f e], and B is left [resp. right] normal if e f g = e g f [resp. e f g = f e g] for every e, f, g∈B. A subset Q of S is a quasi-ideal of S if QSQ ⊆ S.

scholarly journals Naturally ordered regular semigroups with maximum inverses

1989 ◽  
Vol 32 (1) ◽  
pp. 33-39 ◽  
Author(s):  
Tatsuhiko Saito
Keyword(s):  
Regular Semigroup ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inverse Transversal ◽  
Regular Semigroups ◽  
Inverse Subsemigroup ◽  
Necessary And Sufficient

Let S be a regular semigroup. An inverse subsemigroup S° of S is called an inverse transversal if S° contains a unique inverse of each element of S. An inverse transversal S° of S is called multiplicative if x°xyy° is an idempotent of S° for every x, y∈S, where x° denotes the unique inverse of x∈S in S°. In Section 1, we obtain a necessary and sufficient condition in order for inverse transversals to be multiplicative.

scholarly journals A NEW CONSTRUCTION FOR REGULAR SEMIGROUPS WITH QUASI-IDEAL ORTHODOX TRANSVERSALS

2009 ◽  
Vol 86 (2) ◽  
pp. 177-187 ◽  
Author(s):  
XIANGJUN KONG ◽  
XIANZHONG ZHAO
Keyword(s):  
Regular Semigroup ◽  
Structure Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Green’S Relations ◽  
Regular Semigroups ◽  
New Construction ◽  
Necessary And Sufficient ◽  
Equivalent Conditions ◽  
Orthodox Semigroups

AbstractIn any regular semigroup with an orthodox transversal, we define two sets R and L using Green’s relations and give necessary and sufficient conditions for them to be subsemigroups. By using R and L, some equivalent conditions for an orthodox transversal to be a quasi-ideal are obtained. Finally, we give a structure theorem for regular semigroups with quasi-ideal orthodox transversals by two orthodox semigroups R and L.

scholarly journals Associate subgroups of orthodox semigroups

1994 ◽  
Vol 36 (2) ◽  
pp. 163-171 ◽  
Author(s):  
T. S. Blyth ◽  
Emília Giraldes ◽  
M. Paula O. Marques-Smith
Keyword(s):  
Regular Semigroup ◽  
Maximal Subgroup ◽  
Direct Product ◽  
Structure Theorem ◽  
General Situation ◽  
Group Of Units ◽  
Regular Monoid ◽  
Orthodox Semigroups ◽  
Middle Unit

A unit regular semigroup [1, 4] is a regular monoid S such that H1 ∩ A(x) ≠ Ø for every xɛS, where H1, is the group of units and A(x) = {y ɛ S; xyx = x} is the set of associates (or pre-inverses) of x. A uniquely unit regular semigroupis a regular monoid 5 such that |H1 ∩ A(x)| = 1. Here we shall consider a more general situation. Specifically, we consider a regular semigroup S and a subsemigroup T with the property that |T ∩ A(x) = 1 for every x ɛ S. We show that T is necessarily a maximal subgroup Hα for some idempotent α. When Sis orthodox, α is necessarily medial (in the sense that x = xαx for every x ɛ 〈E〉) and αSα is uniquely unit orthodox. When S is orthodox and α is a middle unit (in the sense that xαy = xy for all x, y ɛ S), we obtain a structure theorem which generalises the description given in [2] for uniquely unit orthodox semigroups in terms of a semi-direct product of a band with a identity and a group.

On left quasinormal orthodox semigroups

1983 ◽  
Vol 95 (1-2) ◽  
pp. 59-71 ◽  
Author(s):  
Gracinda M. S. Gomes
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Orthodox Semigroup ◽  
Local Isomorphism ◽  
Orthodox Semigroups

SynopsisThe existence of a smallest inverse congruence on an orthodox semigroup is known. It is also known that a regular semigroup S is locally inverse and orthodox if and only if there exists a local isomorphism from S onto an inverse semigroup T.In this paper, we show the existence of a smallest R-unipotent congruence ρ on an orthodox semigroup S and give its expression in the case where S is also left quasinormal. Finally, we prove that a regular semigroup S is left quasinormal and orthodox if and only if there exists a local isomorphism from S onto an R-unipotent semigroup T.

scholarly journals On weakly multiplicative inverse transversals

1994 ◽  
Vol 37 (1) ◽  
pp. 91-99 ◽  
Author(s):  
T. S. Blyth ◽  
M. H. Almeida Santos
Keyword(s):  
Regular Semigroup ◽  
Inverse Transversal ◽  
Multiplicative Inverse

We show that an inverse transversal of a regular semigroup is multiplicative if and only if it is both weakly multiplicative and a quasi-ideal. Examples of quasi-ideal inverse transversals that are not multiplicative are known. Here we give an example of a weakly multiplicative inverse transversal that is not multiplicative. An interesting feature of this example is that it also serves to show that, in an ordered regular semigroup in which every element x has a biggest inverse x0, the mapping x↦x00 is not in general a closure; nor is x↦x** in a principally ordered regular semigroup.

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