scholarly journals On Epiorthodox Semigroups

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Shouxu Du ◽  
Xinzhai Xu ◽  
K. P. Shum
Keyword(s):  
Structure Theorem ◽  
Orthodox Semigroup ◽  
Normal Band ◽  
Orthodox Semigroups

It has been well known that the band of idempotents of a naturally ordered orthodox semigroup satisfying the “strong Dubreil-Jacotin condition” forms a normal band. In the literature, the naturally ordered orthodox semigroups satisfying the strong Dubreil-Jacotin condition were first considered by Blyth and Almeida Santos in 1992. Based on the name “epigroup” in the paper of Blyth and Almeida Santos and also the name “epigroups” proposed by Shevrin in 1955; we now call the naturally ordered orthodox semigroups satisfying the Dubreil-Jacotin condition theepiorthodox semigroups. Because the structure of this kind of orthodox semigroups has not yet been described, we therefore give a structure theorem for the epi-orthodox semigroups.

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 A note on congruences on orthodox semigroups

1985 ◽  
Vol 26 (1) ◽  
pp. 25-30
Author(s):  
D. B. McAlister
Keyword(s):  
Inverse Semigroup ◽  
Orthodox Semigroup ◽  
Explicit Construction ◽  
Lattice Of Congruences ◽  
Orthodox Semigroups

C. Eberhart and W. Williams [3] showed that the least inverse semigroup congruence , on an orthodox semigroup S, plays a very important role in determining the structure of the lattice of congruences on S. In this note we show that their results can be applied to give an explicit construction for the idempotent separating congruences on S in terms of idempotent separating congruences on S/.

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.

ON E-UNITARY COVERS OF ORTHODOX SEMIGROUPS

1993 ◽  
Vol 03 (03) ◽  
pp. 317-333 ◽  
Author(s):  
MÁRIA B. SZENDREI
Keyword(s):  
Semidirect Product ◽  
Orthodox Semigroup ◽  
Regular Semigroups ◽  
Band Variety ◽  
Orthodox Semigroups

In this paper we prove that each orthodox semigroup S has an E-unitary cover embeddable into a semidirect product of a band B by a group where B belongs to the band variety generated by the band of idempotents in S. This result is related to an embeddability question on E-unitary regular semigroups raised previously.

scholarly journals Semigroups whose idempotents form a subsemigroup

1990 ◽  
Vol 41 (2) ◽  
pp. 161-184 ◽  
Author(s):  
Jean-Camille Birget ◽  
Stuart Margolis ◽  
John Rhodes
Keyword(s):  
Inverse Semigroup ◽  
Finite Semigroup ◽  
Orthodox Semigroup ◽  
Inverse Semigroups ◽  
Type Ii ◽  
Finite Semigroups ◽  
Case Type ◽  
Orthodox Semigroups

We prove that if the “type-II-construct” subsemigroup of a finite semigroup S is regular, then the “type-II” subsemigroup of S is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results:(1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute.(2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup.(3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety u * G if and only if Sn belongs to u. Here u denotes the pseudo-variety of finite semigroups which are unions of groups.For these three classes of semigroups, type-II is equal to type-II construct.

EXTENSIONS OF REGULAR ORTHOGROUPS BY INVERSE SEMIGROUPS

1995 ◽  
Vol 05 (03) ◽  
pp. 317-342 ◽  
Author(s):  
BERND BILLHARDT
Keyword(s):  
Inverse Semigroup ◽  
Recent Result ◽  
Semidirect Product ◽  
Orthodox Semigroup ◽  
Inverse Semigroups ◽  
Normal Extension ◽  
Group Congruence ◽  
Completely Regular ◽  
Orthodox Semigroups

Let V be a variety of regular orthogroups, i.e. completely regular orthodox semigroups whose band of idempotents is regular. Let S be an orthodox semigroup which is a (normal) extension of an orthogroup K from V by an inverse semigroup G, that is, there is a congruence ρ on S such that the semigroup ker ρ of all idempotent related elements of S is isomorphic to K and S/ρ≅G. It is shown that S can be embedded into an orthodox subsemigroup T of a double semidirect product A**G where A belongs to V. Moreover T itself can be chosen to be an extension of a member from V by G. If in addition ρ is a group congruence we obtain a recent result due to M.B. Szendrei [16] which says that each orthodox semigroup which is an extension of a regular orthogroup K by a group G can be embedded into a semidirect product of an orthogroup K′ by G where K′ belongs to the variety of orthogroups generated by K.

scholarly journals Lattice isomorphisms of orthodox semigroups

1992 ◽  
Vol 45 (2) ◽  
pp. 201-213 ◽  
Author(s):  
Katherine G. Johnston ◽  
F.D. Cleary
Keyword(s):  
Orthodox Semigroup ◽  
Inverse Semigroups ◽  
Special Cases ◽  
Orthodox Semigroups

It is shown that the set of all orthodox subsemigroups of an orthodox semigroup forms a lattice. This lattice is a join-sublattice of the lattice of all semigroups, but is not in general a meet-sublattice. We obtain results concerning lattice isomorphisms between orthodox semigroups, several of which include known results for inverse semigroups as special cases.

scholarly journals On regular semigroups whose idempotents form a subsemigroup

1969 ◽  
Vol 1 (2) ◽  
pp. 195-208 ◽  
Author(s):  
T. E. Hall
Keyword(s):  
Inverse Semigroup ◽  
Simple Form ◽  
Finite Semigroup ◽  
Orthodox Semigroup ◽  
Principal Factor ◽  
Regular Semigroups ◽  
Orthodox Semigroups

For brevity the semigroups in the title are called orthodox semigroups. The finest inverse semigroup congruence on an orthodox semigroup is shown to have a simple form and conversely, regular semigroups whose finest inverse congruence has this simple form are shown to be orthodox. Next ideal extensions of orthodox semigroups by orthodox semigroups are shown to be also orthodox, whence a finite semigroup is orthodox if and only if each principal factor is orthodox and completely O-simple or simple. Finally it is determined which completely O-simple semigroups are orthodox.

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.

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