On transitive Cayley graphs of homogeneous inverse semigroups

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.