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 $$P_\mu (X)$$ .