On transitive Cayley graphs of homogeneous inverse semigroups

Abstract

Let S be a pseudo-unitary homogeneous (graded) inverse semigroup with zero 0, that is, an inverse semigroup with zero, and with a family $\{S_\delta\}_{\delta\in\Delta}$ of nonzero subsets of S, called components of S, indexed by a partial groupoid $\Delta$, that is, by a set with a partial binary operation, such that $S=\bigcup_{\delta\in\Delta}S_\delta$, and: i) $S_\xi\cap S_\eta\subseteq\{0\}$ for all distinct $\xi,\eta\in\Delta;$ ii) $S_\xi S_\eta\subseteq S_{\xi\eta}$ whenever $\xi\eta$ is defined; iii) $S_\xi S_\eta\nsubseteq\{0\}$ if and only if the product $\xi\eta$ is defined; iv) for every idempotent element $\epsilon\in\Delta$, the subsemigroup $S_\epsilon$ is with identity $1_\epsilon;$ v) for every $x\in S$ there exist idempotent elements $\xi, \eta\in\Delta$ such that $1_\xi x=x=x1_\eta;$ vi) $1_\xi1_\eta=1_{\xi\eta}$ whenever $\xi\eta\in\Delta$ is an idempotent element, where $\xi$, $\eta$ are idempotent elements of $\Delta$. Let A be a subset of the union of the subsemigroup components of S, which does not contain 0. By $\operatorname{Cay}(S^*,A)$ we denote a graph obtained from the Cayley graph $\operatorname{Cay}(S,A)$ by removing 0 and its incident edges. We characterize vertex-transitivity of $\operatorname{Cay}(S^*,A)$ and relate it to the vertex-transitivity of its subgraph whose vertex set is $S_\mu\setminus\{0\}$, where $\mu$ is the maximum element of the set of all idempotent elements of $\Delta$, with respect to the natural order.