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

Allen Zhang

TL;DR

This work develops an algebraic framework for partial actions of groups on generalized Boolean algebras tied to strongly $E^{\ast}$-unitary inverse semigroups. By constructing a partial action on the tight-idempotent Boolean algebra and forming the skew group ring $L_R(S, \varphi) = \text{Lc}(R, \mathcal{B}) \rtimes_{\Phi} G$, the authors obtain an isomorphism with the Steinberg algebra of the tight groupoid, thereby connecting inverse semigroup theory, generalized Boolean algebras, and groupoid algebras. A key contribution is showing that this construction is independent of the chosen pure grading via the universal group, and that unitization on the inverse semigroup corresponds to unitization on the associated $R$-algebra. The theory is then applied to realize Leavitt path algebras and labelled Leavitt path algebras as the $R$-algebras attached to strongly $E^{\ast}$-unitary inverse semigroups, unifying several combinatorial-algebraic constructions under a common skew-group-ring paradigm.

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.

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

TL;DR

This work develops an algebraic framework for partial actions of groups on generalized Boolean algebras tied to strongly -unitary inverse semigroups. By constructing a partial action on the tight-idempotent Boolean algebra and forming the skew group ring , the authors obtain an isomorphism with the Steinberg algebra of the tight groupoid, thereby connecting inverse semigroup theory, generalized Boolean algebras, and groupoid algebras. A key contribution is showing that this construction is independent of the chosen pure grading via the universal group, and that unitization on the inverse semigroup corresponds to unitization on the associated -algebra. The theory is then applied to realize Leavitt path algebras and labelled Leavitt path algebras as the -algebras attached to strongly -unitary inverse semigroups, unifying several combinatorial-algebraic constructions under a common skew-group-ring paradigm.

Abstract

We define the notion of a partial action on a generalized Boolean algebra and associate to every such system and commutative unital ring an -algebra. We prove that every strongly -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 -unitary inverse semigroup and commutative unital ring an -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 -algebra. Finally, we show that Leavitt path algebras and labelled Leavitt path algebras can be realized as the -algebra associated to a strongly -unitary inverse semigroup.

Paper Structure

This paper contains 14 sections, 43 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Partial Actions on Generalized Boolean Algebras and an Associated Skew Group Ring
  3. Generalized Boolean Algebras and Partial Actions
  4. An Associated Partial Skew Group Ring
  5. Meet Lattices and Tight Filters
  6. Meet Semilattices and Filters
  7. Maps on Tight Filters Induced by Meet Subsemilattices
  8. Partial Actions from Strongly $E^{\ast}$-Unitary Inverse Semigroups
  9. Basic Definitions
  10. Building a Partial Action
  11. Unitization
  12. Proofs of Isomorphism
  13. Leavitt Path Algebras
  14. Labelled Leavitt Path Algebras

Key Result

Lemma 2.16

Let $X(\mathcal{B})$ be the Stone dual of $\mathcal{B}$. Then $\mathrm{Lc}(R, \mathcal{B})$ is isomorphic as an $R$-algebra to the set of locally constant functions with compact support from $X(\mathcal{B}) \rightarrow R$ where $R$ has the discrete topology.

Theorems & Definitions (130)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Remark 2.7
  • Definition 2.8
  • Example 2.9
  • Definition 2.10
  • ...and 120 more