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
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