Semigroups of partial transformations with kernel and image restricted by an equivalence

Abstract

For an arbitrary set X and an equivalence relation $\mu$ on X, denote by $P_\mu (X)$ the semigroup of partial transformations $\alpha$ on X such that $x\mu \subseteq x(\ker (\alpha ))$ for every $x\in {{\,\mathrm{dom}\,}}(\alpha )$, and the image of $\alpha$ is a partial transversal of $\mu$. Every transversal K of $\mu$ defines a subgroup $G=G_{K}$ of $P_\mu (X)$. We study subsemigroups $\langle G,U\rangle$ of $P_\mu (X)$ generated by $G\cup U$, where U is any set of elements of $P_\mu (X)$ of rank less than $|X/\mu |$. We show that each $\langle G,U\rangle$ is a regular semigroup, describe Green’s relations and ideals in $\langle G,U\rangle$, and determine when $\langle G,U\rangle$ is an inverse semigroup and when it is a completely regular semigroup. For a finite set X, the top $\mathcal {J}$-class J of $P_\mu (X)$ is a right group. We find formulas for the ranks of the semigroups J, $G\cup I$, $J\cup I$, and I, where I is any proper ideal of $P_\mu (X)$.