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- W3039609822 abstract "Let $X$ be a nonempty set, and let $mathcal{T}_X$ be the full transformation semigroup on $X$. For a partition $mathcal{P} = {X_i ;|; iin I}$ of $X$, we consider the semigroup $T(X, mathcal{P}) = {fin mathcal{T}_X;|; forall X_i;exists X_j,; X_i f subseteq X_j}$, the subsemigroup $Sigma(X, mathcal{P}) = {fin T(X, mathcal{P});|; Xf cap X_i neq emptyset; forall X_i}$, and the group of units $S(X, mathcal{P})$ of $T(X, mathcal{P})$. In this paper, we first characterize the elements of $Sigma(X, mathcal{P})$. For a permutation $f$ of finite $X$, we next observe whether there exists a nontrivial partition $mathcal{P}$ of $X$ such that $fin S(X, mathcal{P})$. We then characterize and enumerate the idempotents in the semigroup $Sigma(X, mathcal{P})$ for arbitrary and finite $X$, respectively. We also characterize the elements of $S(X, mathcal{P})$. For finite $X$, we finally calculate the cardinality of $T(X, mathcal{P})$, $Sigma(X, mathcal{P})$, and $S(X, mathcal{P})$." @default.
- W3039609822 created "2020-07-10" @default.
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- W3039609822 date "2020-09-13" @default.
- W3039609822 modified "2023-09-25" @default.
- W3039609822 title "On certain semigroups of transformations that preserve a partition" @default.
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- W3039609822 doi "https://doi.org/10.1080/00927872.2020.1799224" @default.
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