Jehan Al-Bar - Academia.edu (original) (raw)

Papers by Jehan Al-Bar

Ars Mathematica Contemporanea, 2017

We analyse the normal quotient structure of several infinite families of finite connected edge-tr... more We analyse the normal quotient structure of several infinite families of finite connected edge-transitive, four-valent oriented graphs. These families were singled out by Marušič and others to illustrate various different internal structures for these graphs in terms of their alternating cycles (cycles in which consecutive edges have opposite orientations). Studying the normal quotients gives fresh insights into these oriented graphs: in particular we discovered some unexpected 'cross-overs' between these graph families when we formed normal quotients. We determine which of these oriented graphs are 'basic', in the sense that their only proper normal quotients are degenerate. Moreover, we show that the three types of edge-orientations studied are the only orientations, of the underlying undirected graphs in these families, which are invariant under a group action which is both vertex-transitive and edge-transitive.

Semigroup Forum, 2010

We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant sem... more We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant semigroups in terms of spined products, which is similar in nature to that given by Saito for weakly multiplicative inverse transversals of regular semigroups [10]. As a consequence we deduce a similar result for multiplicative transversals of abundant semigroups and also consider the case when the semigroups are in fact regular and provide some new structure theorems for inverse transversals.

Communications in Algebra, 2012

The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [... more The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products of adequate semigroups by left regular bands.

We consider adequate transversals of abundant semigroups and prove that, in a partic-ular case, t... more We consider adequate transversals of abundant semigroups and prove that, in a partic-ular case, there is a natural embedding of an inverse transversal within a certain regular subsemigroup. We also introduce the concepts of simplistic, perfect and quasi-adequate transversal and provide a number of interesting connections between these. 1

We develop a new framework for analysing finite connected, oriented graphs of valency 4, which ad... more We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of "basic" graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restirictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.

Semigroup Forum, Aug 31, 2011

Semigroup Forum, 2010

We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant sem... more We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant semigroups in terms of spined products, which is similar in nature to that given by Saito in Proc. 8th Symposium on Semigroups, pp. 22–25 (1985) for weakly multiplicative inverse transversals of regular semigroups. As a consequence we deduce a similar result for multiplicative transversals of abundant semigroups and also consider the case when the semigroups are in fact regular and provide some new structure theorems for inverse transversals.

We consider adequate transversals of abundant semigroups and prove that, in a particular case, th... more We consider adequate transversals of abundant semigroups and prove that, in a particular case, there is a natural embedding of an inverse transversal within a certain regular subsemigroup. We also introduce the concepts of simplistic, perfect and quasi-adequate transversal and provide a number of interesting connections between these.

Communications in Algebra, 2009

We consider adequate transversals of abundant semigroups and prove that, in a particular case, th... more We consider adequate transversals of abundant semigroups and prove that, in a particular case, there is a natural embedding of an inverse transversal within a certain regular subsemigroup. We also introduce the concepts of simplistic, perfect and quasi-adequate transversal and provide a number of interesting connections between these.

The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [... more The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products of adequate semigroups by left regular bands.

Ars Mathematica Contemporanea, 2017

We analyse the normal quotient structure of several infinite families of finite connected edge-tr... more We analyse the normal quotient structure of several infinite families of finite connected edge-transitive, four-valent oriented graphs. These families were singled out by Marušič and others to illustrate various different internal structures for these graphs in terms of their alternating cycles (cycles in which consecutive edges have opposite orientations). Studying the normal quotients gives fresh insights into these oriented graphs: in particular we discovered some unexpected 'cross-overs' between these graph families when we formed normal quotients. We determine which of these oriented graphs are 'basic', in the sense that their only proper normal quotients are degenerate. Moreover, we show that the three types of edge-orientations studied are the only orientations, of the underlying undirected graphs in these families, which are invariant under a group action which is both vertex-transitive and edge-transitive.

Semigroup Forum, 2010

We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant sem... more We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant semigroups in terms of spined products, which is similar in nature to that given by Saito for weakly multiplicative inverse transversals of regular semigroups [10]. As a consequence we deduce a similar result for multiplicative transversals of abundant semigroups and also consider the case when the semigroups are in fact regular and provide some new structure theorems for inverse transversals.

Communications in Algebra, 2012

The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [... more The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products of adequate semigroups by left regular bands.

We consider adequate transversals of abundant semigroups and prove that, in a partic-ular case, t... more We consider adequate transversals of abundant semigroups and prove that, in a partic-ular case, there is a natural embedding of an inverse transversal within a certain regular subsemigroup. We also introduce the concepts of simplistic, perfect and quasi-adequate transversal and provide a number of interesting connections between these. 1

We develop a new framework for analysing finite connected, oriented graphs of valency 4, which ad... more We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of "basic" graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restirictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.

Semigroup Forum, Aug 31, 2011

Semigroup Forum, 2010

We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant sem... more We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant semigroups in terms of spined products, which is similar in nature to that given by Saito in Proc. 8th Symposium on Semigroups, pp. 22–25 (1985) for weakly multiplicative inverse transversals of regular semigroups. As a consequence we deduce a similar result for multiplicative transversals of abundant semigroups and also consider the case when the semigroups are in fact regular and provide some new structure theorems for inverse transversals.

We consider adequate transversals of abundant semigroups and prove that, in a particular case, th... more We consider adequate transversals of abundant semigroups and prove that, in a particular case, there is a natural embedding of an inverse transversal within a certain regular subsemigroup. We also introduce the concepts of simplistic, perfect and quasi-adequate transversal and provide a number of interesting connections between these.

Communications in Algebra, 2009

We consider adequate transversals of abundant semigroups and prove that, in a particular case, th... more We consider adequate transversals of abundant semigroups and prove that, in a particular case, there is a natural embedding of an inverse transversal within a certain regular subsemigroup. We also introduce the concepts of simplistic, perfect and quasi-adequate transversal and provide a number of interesting connections between these.

The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [... more The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products of adequate semigroups by left regular bands.