scholarly journals Cross-connection structure of concordant semigroups

2019 ◽  
Vol 30 (01) ◽  
pp. 181-216
Author(s):  
P. A. Azeef Muhammed ◽  
P. G. Romeo ◽  
K. S. S. Nambooripad
Keyword(s):  
Ideal Structure ◽  
Abundant Semigroup ◽  
Regular Semigroups ◽  
Category Equivalence ◽  
The Cross ◽  
Green Relations ◽  
Cross Connection

Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalized Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the category of concordant semigroups and the category of cross-connected consistent categories. In the process, we illustrate how our construction is a generalization of the cross-connection analysis of regular semigroups. We also identify the inductive cancellative category associated with a pair of cross-connected consistent categories.

A Structure Theorem of Regular ${\cal H}^\sharp$-Cryptographs

2008 ◽  
Vol 15 (04) ◽  
pp. 653-666 ◽  
Author(s):  
Xiangzhi Kong ◽  
Zhiling Yuan ◽  
K. P. Shum
Keyword(s):  
Structure Theorem ◽  
Abundant Semigroup ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Abundant Semigroups ◽  
Completely Regular Semigroups ◽  
Semilattice Of Semigroups ◽  
Strong Semilattice ◽  
Green Relations ◽  
Strong Semilattice Of Semigroups

A new set of generalized Green relations is given in studying the [Formula: see text]-abundant semigroups. By using the generalized strong semilattice of semigroups recently developed by the authors, we show that an [Formula: see text]-abundant semigroup is a regular [Formula: see text]-cryptograph if and only if it is an [Formula: see text]-strong semilattice of completely [Formula: see text]-simple semigroups. This result not only extends the well known result of Petrich and Reilly from the class of completely regular semigroups to the class of semiabundant semigroups, but also generalizes a well known result of Fountain on superabundant semigroups from the class of abundant semigroups to the class of semiabundant semigroups.

scholarly journals Cross-connections of the singular transformation semigroup

2018 ◽  
Vol 17 (03) ◽  
pp. 1850047 ◽  
Author(s):  
P. A. Azeef Muhammed ◽  
A. R. Rajan
Keyword(s):  
General Theory ◽  
Transformation Semigroup ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Regular Semigroups ◽  
Partially Ordered ◽  
Singular Transformation ◽  
The Cross ◽  
The Right ◽  
Cross Connection

Cross-connection is a construction of regular semigroups using certain categories called normal categories which are abstractions of the partially ordered sets of principal left (right) ideals of a semigroup. We describe the cross-connections in the semigroup [Formula: see text] of all non-invertible transformations on a set [Formula: see text]. The categories involved are characterized as the powerset category [Formula: see text] and the category of partitions [Formula: see text]. We describe these categories and show how a permutation on [Formula: see text] gives rise to a cross-connection. Further, we prove that every cross-connection between them is induced by a permutation and construct the regular semigroups that arise from the cross-connections. We show that each of the cross-connection semigroups arising this way is isomorphic to [Formula: see text]. We also describe the right reductive subsemigroups of [Formula: see text] with the category of principal left ideals isomorphic to [Formula: see text]. This study sheds light into the more general theory of cross-connections and also provides an alternate way of studying the structure of [Formula: see text].

scholarly journals The characterization and the product of quasi-Ehresmann transversals

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2051-2060
Author(s):  
Xiangjun Kong ◽  
Pei Wang
Keyword(s):  
Abundant Semigroup ◽  
Regular Semigroups ◽  
The Real ◽  
Abundant Semigroups

Wang (Filomat 29(5), 985-1005, 2015) introduced and investigated quasi-Ehresmann transversals of semi-abundant semigroups satisfy conditions (CR) and (CL) as the generalizations of orthodox transversals of regular semigroups in the semi-abundant case. In this paper, we give two characterizations for a generalized quasi-Ehresmann transversal to be a quasi-Ehresmann transversal. These results further demonstrate that quasi-Ehresmann transversals are the ?real? generalizations of orthodox transversals in the semi-abundant case. Moreover, we obtain the main result that the product of any two quasi-ideal quasi-Ehresmann transversals of a semi-abundant semigroup S which satisfy the certain conditions is a quasi-ideal quasi-Ehresmann transversal of S.

scholarly journals Cross-connections of completely simple semigroups

2016 ◽  
Vol 09 (03) ◽  
pp. 1650053 ◽  
Author(s):  
P. A. Azeef Muhammed ◽  
A. R. Rajan
Keyword(s):  
Simple Semigroup ◽  
Structure Theory ◽  
Cambridge Philos ◽  
Regular Semigroups ◽  
Local Isomorphism ◽  
Completely Simple Semigroup ◽  
Matrix Formula ◽  
Matrix Semigroup ◽  
Completely Simple ◽  
Cross Connection

A completely simple semigroup [Formula: see text] is a semigroup without zero which has no proper ideals and contains a primitive idempotent. It is known that [Formula: see text] is a regular semigroup and any completely simple semigroup is isomorphic to the Rees matrix semigroup [Formula: see text] (cf. D. Rees, On semigroups, Proc. Cambridge Philos. Soc. 36 (1940) 387–400). In the study of structure theory of regular semigroups, Nambooripad introduced the concept of normal categories to construct the semigroup from its principal left (right) ideals using cross-connections. A normal category [Formula: see text] is a small category with subobjects wherein each object of the category has an associated idempotent normal cone and each morphism admits a normal factorization. A cross-connection between two normal categories [Formula: see text] and [Formula: see text] is a local isomorphism [Formula: see text] where [Formula: see text] is the normal dual of the category [Formula: see text]. In this paper, we identify the normal categories associated with a completely simple semigroup [Formula: see text] and show that the semigroup of normal cones [Formula: see text] is isomorphic to a semi-direct product [Formula: see text]. We characterize the cross-connections in this case and show that each sandwich matrix [Formula: see text] correspond to a cross-connection. Further we use each of these cross-connections to give a representation of the completely simple semigroup as a cross-connection semigroup.

scholarly journals Self-injectivity of semigroup algebras

2020 ◽  
Vol 18 (1) ◽  
pp. 333-352
Author(s):  
Junying Guo ◽  
Xiaojiang Guo
Keyword(s):  
Regular Semigroup ◽  
Necessary Conditions ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Abundant Semigroup ◽  
Regular Semigroups ◽  
Abundant Semigroups ◽  
Sufficient And Necessary Conditions ◽  
Finite Regular Semigroup

Abstract It is proved that for an IC abundant semigroup (a primitive abundant semigroup; a primitively semisimple semigroup) S and a field K, if K 0[S] is right (left) self-injective, then S is a finite regular semigroup. This extends and enriches the related results of Okniński on self-injective algebras of regular semigroups, and affirmatively answers Okniński’s problem: does that a semigroup algebra K[S] is a right (respectively, left) self-injective imply that S is finite? (Semigroup Algebras, Marcel Dekker, 1990), for IC abundant semigroups (primitively semisimple semigroups; primitive abundant semigroups). Moreover, we determine the structure of K 0[S] being right (left) self-injective when K 0[S] has a unity. As their applications, we determine some sufficient and necessary conditions for the algebra of an IC abundant semigroup (a primitively semisimple semigroup; a primitive abundant semigroup) over a field to be semisimple.

REES MATRIX COVERS FOR TIGHT ABUNDANT SEMIGROUPS

2010 ◽  
Vol 03 (03) ◽  
pp. 409-425
Author(s):  
Xiaojiang Guo ◽  
K. P. Shum ◽  
Yongqian Zhu
Keyword(s):  
Rees Matrix Semigroup ◽  
Abundant Semigroup ◽  
Regular Semigroups ◽  
Regular Elements ◽  
Abundant Semigroups ◽  
Matrix Semigroup

Rees matrix covers for regular semigroups were first studied by McAlister in 1984. Lawson extended McAlister's results to abundant semigroups in 1987. We consider here a semigroup whose set of regular elements forms a subsemigroup, named tight semigroups. In this paper, it is proved that an abundant semigroup is tight and locally E-solid if and only if it is an F-local isomorphic image of an abundant Rees matrix semigroup [Formula: see text] over a tight E-solid abundant semigroup T, where the entries of the sandwich matrix P of [Formula: see text] are regular elements of T. Our results enrich the result of Lawson on Rees matrix covers for a class of abundant semigroups and extend the results of McAlister on Rees matrix covers for regular semigroups.

Semiprimeness of semigroup algebras

2021 ◽  
Vol 19 (1) ◽  
pp. 803-832
Author(s):  
Junying Guo ◽  
Xiaojiang Guo
Keyword(s):  
Regular Semigroup ◽  
Regularity Condition ◽  
Necessary Conditions ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Abundant Semigroup ◽  
Locally Finite ◽  
Regular Semigroups ◽  
Abundant Semigroups ◽  
Sufficient And Necessary Conditions

Abstract Abundant semigroups originate from p.p. rings and are generalizations of regular semigroups. The main aim of this paper is to study the primeness and the primitivity of abundant semigroup algebras. We introduce and study D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices of a semigroup. Based on D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices, we determine when a contracted semigroup algebra of a primitive abundant semigroup is prime (respectively, semiprime, semiprimitive, or primitive). It is well known that for any algebra A {\mathcal{A}} with unity, A {\mathcal{A}} is primitive (prime) if and only if so is M n ( A ) {M}_{n}\left({\mathcal{A}}) . Our results can be viewed as some kind of generalizations of such a known result. In addition, it is proved that any contracted semigroup algebra of a locally ample semigroup whose set of idempotents is locally finite (respectively, locally pseudofinite and satisfying the regularity condition) is isomorphic to some contracted semigroup algebra of primitive abundant semigroups. Moreover, we obtain sufficient and necessary conditions for these classes of contracted semigroup algebras to be prime (respectively, semiprime, semiprimitive, or primitive). Finally, the structure of simple contracted semigroup algebras of idempotent-connected abundant semigroups is established. Our results enrich and extend the related results on regular semigroup algebras.

scholarly journals The natural partial order on an abundant semigroup

1987 ◽  
Vol 30 (2) ◽  
pp. 169-186 ◽  
Author(s):  
Mark V. Lawson
Keyword(s):  
Partial Order ◽  
Natural Partial Order ◽  
Abundant Semigroup ◽  
Regular Semigroups

In this paper we will study the properties of a natural partial order which may bedefined on an arbitrary abundant semigroup: in the case of regular semigroups werecapture the order introduced by Nambooripad [24]. For abelian PP rings our order coincides with a relation introduced by Sussman [25], Abian [1, 2] and further studied by Chacron [7]. Burmistroviˇ [6] investigated Sussman's order on separative semigroups. In the abundant case his order coincides with ours: some order theoretic properties of such semigroups may be found in a paper by Burgess [5].

Analysis of postural motoneuron activity in crayfish abdomen. II. Coordination by excitatory and inhibitory connections between motoneurons

1975 ◽  
Vol 38 (2) ◽  
pp. 332-346 ◽  
Author(s):  
W. G. Tatton ◽  
P. G. Sokolove
Keyword(s):  
Cross Correlation ◽  
Spike Trains ◽  
Efferent Activity ◽  
Efferent Neurons ◽  
Postural System ◽  
Motoneuron Activity ◽  
The Cross ◽  
Cross Connection

The identified spike trains of the crayfish abdominal postural efferent neurons were recorded simultaneously from one or more segments. The efferent activity was analyzed using cross-correlation histograms, peristimulus time scatter diagrams, and specialized antidromic techniques. The analyses show that the larger, phasically active motoneurons are coordinated in their activity by cross connections made at the motoneuron level. The cross connections are both excitatory and inhibitor in nature and result in significant alterations in spike output. Further, the accessory neuron receives an inhibitory cross connection from a middle-sized extensor excitor motoneuron or motoneurons. In each case, the cross connections appear to be appropriate to the function of the postural system.

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