On certain semigroups of transformations that preserve a partition
Mosarof Sarkar, Shubh N. Singh · Communications in Algebra · 2020
Let X be a nonempty set, and let TX be the full transformation semigroup on X. For a partition P={Xi|i∈I} of X, we consider the semigroup T(X,P)={f∈TX|∀Xi∃Xj,Xif⊆Xj}, the subsemigroup Σ(X,P)={f∈T(X,P)|Xf∩Xi≠∅∀Xi}, and the group of units S(X,P) of T(X,P). In this paper, we first characterize the elements of Σ(X,P). For a permutation f of finite X, we next observe whether there exists a nontrivial partition P of X such that f∈S(X,P). We then characterize and enumerate the idempotents in the semigroup Σ(X,P) for arbitrary and finite X, respectively. We also characterize the elements of S(X,P). For finite X, we finally calculate the cardinality of T(X,P),Σ(X,P), and S(X,P).