### E-Inversive Elements in Some Semigroups of Transformations that Preserve Equivalence

#### Abstract

Let $X$ be a nonempty set and $T(X)$ the full transformation semigroup on a set $X$. For an equivalence relation $E$ on $X$ and a cross-section $R$ of the partition $X/E$ induced by $E$, let \[ T_{E^{*}}(X) = \{\alpha \in T(X) : \forall x, y \in X, (x, y) \in E \Leftrightarrow (x\alpha, y\alpha) \in E \} \text{ and } \] \[ T_{E}(X, R) = \{\alpha \in T(X) : R\alpha = R \mbox{ and } \forall x,y \in X, (x, y) \in E \Rightarrow (x\alpha, y\alpha) \in E \}. \] Then $T_{E^{*}}(X)$ and $T_{E}(X, R)$ are subsemigroups of $T(X)$. In this paper, we describe the $E$-inversive elements of $T_{E^{*}}(X)$ and $T_{E}(X, R)$. We also show that $T_{E^{*}}(X)$ and $T_{E}(X, R)$ are $E$-inversive semigroups in terms of the cardinality of $X/E$ and $R$, respectively.

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