Partial actions on generalized Boolean algebras with applications to inverse semigroups and combinatorial R-algebras
Year: 2026
Type: article
Abstract: In this paper, we define the notion of a partial action on a generalized Boolean algebra and associate to every such system and commutative unital ring [Formula: see text] an [Formula: see text]-algebra. We prove that every strongly [Formula: see text]-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 [Formula: see text]-unitary inverse semigroup and commutative unital ring [Formula: see text] an [Formula: see text]-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 [Formula: see text]-algebra. Finally, we show that Leavitt path algebras and labeled Leavitt path algebras can be realized as the [Formula: see text]-algebra associated to a strongly [Formula: see text]-unitary inverse semigroup.
Author: Allen Zhang
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7169518509
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