Partial Actions on Generalized Boolean Algebras with Applications to Inverse Semigroups and Combinatorial $R$-Algebras

Abstract: We define the notion of a partial action on a generalized Boolean algebra and associate to every such system and commutative unital ring $R$ an $R$-algebra. We prove that every strongly $E^{\ast}$-unitary inverse semigroup has an associated partial action on the generalized Boolean algebra of compact open sets of tight filters in the meet semilattice of idempotents. Using these correspondences, we associate to every strongly $E^{\ast}$-unitary inverse semigroup and commutative unital ring $R$ an $R$-algebra, and show that it is isomorphic to the Steinberg algebra of the tight groupoid. As an application, we show that there is a natural unitization operation on an inverse semigroup that corresponds to a unitization of the corresponding $R$-algebra. Finally, we show that Leavitt path algebras and labelled Leavitt path algebras can be realized as the $R$-algebra associated to a strongly $E^{\ast}$-unitary inverse semigroup.