Abstract
Let X be a nonempty set, and let be the full transformation semigroup on X. For a partition
of X, we consider the semigroup
the subsemigroup
and the group of units
of
In this paper, we first characterize the elements of
For a permutation f of finite X, we next observe whether there exists a nontrivial partition
of X such that
We then characterize and enumerate the idempotents in the semigroup
for arbitrary and finite X, respectively. We also characterize the elements of
For finite X, we finally calculate the cardinality of
and
Acknowledgment
We are thankful to the referee for bringing the paper [Citation1] to our notice regarding results in the section 4, and helpful suggestions to improve the manuscript.