2,603 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
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