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      Some Results on certain Semigroups of Partition-Preserving Transformations

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          Abstract

          Let \(T_X\) and \(S_X\) be the full transformation semigroup and the symmetric group on a nonempty set \(X\), respectively. For a partition \(\mathcal{P} = \{X_i |\; i\in I\}\) of a nonempty set \(X\) indexed by the set \(I\), we study some aspects of the semigroups \(T(X, \mathcal{P}) = \{f\in T_X|\; \forall X_i\;\exists X_j,\; X_i f \subseteq X_j\}\), \(\Sigma(X, \mathcal{P}) = \{f\in T(X, \mathcal{P})|\; Xf \cap X_i \neq \emptyset\; \forall X_i\}\), and the subgroup \(S(X, \mathcal{P}) = T(X, \mathcal{P}) \cap S_X\). In fact, we first characterize the mapping \(\chi^{(f)}\colon I \to I\), corresponding to a mapping \(f\in T(X, \mathcal{P})\), defined by setting \(i \chi^{(f)} = j\) whenever \(X_if \subseteq X_j\). We next find a partial affirmative answer to the natural question: For \(f\in T_X(f\in S_X)\), whether \(f\in T(X, \mathcal{P})\;(f\in S(X, \mathcal{P}))\) for some nontrivial partition \(\mathcal{P}\) of a finite set \(X\)? We further give a necessary and sufficient condition for a mapping of \(T(X, \mathcal{P})\) to be in \(S(X, \mathcal{P})\). We also obtain a formula for the idempotent elements in the finite semigroup \(\Sigma(X, \mathcal{P})\). We finally determine the sizes of the finite semigroups \(T(X, \mathcal{P})\), \(\Sigma(X, \mathcal{P})\), and the finite subgroup \(S(X, \mathcal{P})\).

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          Author and article information

          Journal
          07 June 2020
          Article
          2006.04242
          c70b1b75-0d3c-4039-ad83-a16f9b4e0836

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          20M15, 20M20
          math.GR

          Algebra
          Algebra

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