On the structure and skeletals of principal ideal graphs of inverse semigroups

The principal left ideal graph of a semigroup is a simple graph whose vertices are the non-zero elements of the semigroup, and two vertices are adjacent if their principal left ideals intersect non-trivially. In this paper, we study the structure of the principal ideal graphs of inverse semigroups, particularly symmetric inverse semigroups. We also introduce the concept of skeletal of a graph and show that the principal ideal graph of an inverse semigroup has a skeletal, which is a simple graph with vertex set as $\mathcal{L}$ classes of non-zero elements. It is also proved that the principal ideal graph of symmetric inverse semigroups has a skeletal which is isomorphic to the intersection graph on the power set of a non-empty set.