12 research outputs found
Cross-connections and variants of the full transformation semigroup
Cross-connection theory propounded by K. S. S. Nambooripad describes the
ideal structure of a regular semigroup using the categories of principal left
(right) ideals. A variant of the full transformation
semigroup for an arbitrary
is the semigroup with the binary
operation where . In this article, we describe the ideal structure of
the regular part of the variant of the full
transformation semigroup using cross-connections. We characterize the
constituent categories of and describe how they are
\emph{cross-connected} by a functor induced by the sandwich transformation
. This lead us to a structure theorem for the semigroup and give the
representation of as a cross-connection semigroup.
Using this, we give a description of the biordered set and the sandwich sets of
the semigroup
Cross-connections of the singular transformation semigroup
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 of all non-invertible
transformations on a set . The categories involved are characterized as the
powerset category and the category of partitions . We
describe these categories and show how a permutation on 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 . We also describe the right
reductive subsemigroups of with the category of principal left ideals
isomorphic to . This study sheds light into the more general
theory of cross-connections and also provides an alternate way of studying the
structure of
Cross-connection structure of concordant semigroups
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 generalised 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
generalisation of Nambooripad's cross-connection analysis of regular
semigroups. We also identify the inductive cancellative category associated
with a pair of cross-connected consistent categories
A Tale of Two Categories: Inductive Groupoids and Cross-connections
A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialized groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialized small category whose object set is a strict preorder and the morphisms admit a factorization property. A pair of ‘related’ normal categories constitutes a cross-connection. Both inductive groupoids and cross-connections were identified by Nambooripad as categorical models of regular semigroups. We explore the inter-relationship between these seemingly different categorical structures and prove a direct category equivalence between the category of inductive groupoids and the category of cross-connections. © 2021 Elsevier B.V.We acknowledge the financial support by the Ministry of Science and Higher Education of the Russian Federation (Ural Mathematical Center project No. 075-02-2021-1387 )
Cross-connection structure of concordant semigroups
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. © 2020 World Scientific Publishing Company.The first author acknowledges the financial support of the Competitiveness Enhancement Program of Ural Federal University, Russia during the preparation of this paper