Certain Local Subsemigroups of Semigroups of Linear Transformations
Keywords:
Local subsemigroup, semigroup of linear transformationsAbstract
A local subsemigroup of a semigroup is a subsemigroup of
of the form
where
is a subsemigroup of
and
is an idempotent of
. It has been shown that for a finite nonempty set
and an idempotent
of
,
is a local subsemigroup of
if and only if either
is the identity mapping on
or for every
,
where
and
are the full transformation semigroup and the symmetric group on
, respectively. In this paper, a parallel result is provided on the semigroup
, under composition, of all linear transformations of a vector space
. We show that for a finite-dimensional vector space
and an idempotent
of
,
is a local subsemigroup of
if and only if either
is the identity mapping on
or
where
is the group of isomorphisms of
.
