Variant of finite symmetric inverse semigroup

M. Balarabe, G. U. Garba, A. T. Imam


In a semigroup $S$ fixed an element $a\in S$ and, for all $x,y\in S$, define a binary operation $*_{a}$ on $S$ by $x*_{a}y=xay$, (where the juxterposition on the left denote the original semigroup operation on $S$). The operation $*_{a}$ is clearly associative and so $S$ forms a new semigroup under this operation, which is denoted by $S^{a}$ and called {\emph{variant}} of $S$ by $a$. For a finite set $X_{n} = \{1, 2,\ldots, n\}$, let $\mathcal{I}_{n}$ be the symmetric inverse semigroup on $X_{n}$ and fix an idempotent $a\in \mathcal{I}_{n}$. In this paper, we study the variant $\mathcal{I}^a_{n}$ of $\mathcal{I}_{n}$ by $a$. In particular, we characterised Green's relations and starred Green's relations $\mathcal{L}^{*}$, $\mathcal{R}^{*}$ in $\mathcal{I}^a_{n}$ and also showed that the variant semigroup $\mathcal{I}^a_{n}$ is abundant.

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