Constructive Stone representations for separated swap and Boolean algebras

Abstract: Swap algebras generalise Bishop's complemented powerset as Boolean algebras generalise the powerset. Actually, all Boolean algebras are swap algebras. We prove constructively a Stone representation theorem for separated swap algebras of type (II), where the notion of a separated swap algebra generalises the corresponding notion of a separated Boolean algebra. Moreover, we prove a Stone-Cech theorem for swap algebras of type (II), showing that the restriction to separated swap algebras is not a loss of generality from the point of view of the theory of swap characters. A constructive Stone representation theorem and a Stone-Cech theorem for Boolean algebras follow as special cases. We introduce sets with a Boolean inequality, that is sets with an internal falsum. These sets allow a book-keeping of the use of the Ex falso principle in constructive mathematics. If we restrict to swap algebras with a Boolean inequality, then the proof of the Stone representation theorem for swap algebras of type (II) is within minimal logic.

• The paper presents a constructive Stone representation theorem for separated swap algebras of type (II) and Boolean algebras, eliminating reliance on classical non-constructive principles.
• It introduces a framework based on Bishop Set Theory and extensional properties to rigorously define complemented subsets and partial Boolean-valued functions.
• The results establish constructive duality and provide categorical tools for extending representation theorems to topological and algebraic structures in minimal logic.

Constructive Stone Representation Theorems for Separated Swap and Boolean Algebras

Introduction

The paper "Constructive Stone representations for separated swap and Boolean algebras" (2606.08176) investigates the constructive analogues of classical Stone representation theorems, focusing on swap algebras, their relation to Bishop-style constructive mathematics (BISH), and extensions to Boolean algebras. The authors develop constructive representation theorems for separated swap algebras of type $(II)$ and analyze their implications for the theory of Boolean algebras. They further address foundational issues of equality, inequality, and the avoidance of non-constructive principles (specifically, Ex falso quodlibet and choice axioms) by introducing sets endowed with Boolean inequality.

Framework: Bishop Set Theory and Constructive Algebra

The foundational setting is Bishop Set Theory (BST), which permits a constructive treatment of sets, assignment routines, and algebraic structures without reliance on classical choice principles. BST is interpreted in type theory via setoids and is compatible with other constructive formal systems, e.g., Myhill's Constructive Set Theory and Aczel's $CZF$.

Swap algebras generalize complemented powersets (as studied constructively by Bishop and others) and abstract complemented subsets, yielding an algebraic framework that subsumes Boolean algebras. The paper distinguishes types $(I)$ and $(II)$ swap algebras based on the operational core of join and meet definitions, noting that all Boolean algebras are swap algebras of both types. Constructive duality is established between swap algebras and swap rings, extending classical duality between Boolean algebras and Boolean rings.

Constructivization Challenges and Solutions

Two principal obstacles impede constructive Stone-type representation for Boolean algebras:

1. Use of Points: Classical pointed Boolean characters require the Law of Excluded Middle (PEM) for their definition. Constructively, the authors circumvent PEM by employing swap characters defined on complemented subsets, with domain restricted to those containing the given point, thus producing pointed swap characters for separated swap algebras of type $(II)$ constructively.
2. Separation and Choice Principles: The classical guarantee that every Boolean algebra is separated depends on Zorn's Lemma or the Boolean Prime Ideal Theorem. The authors show that restricting to separated swap or Boolean algebras is not a loss of generality, via a constructive Stone–Čech theorem for swap algebras of type $(II)$, which ensures that for any swap algebra, its swap characters correspond to those in a separated swap algebra.

Formal Development: Sets with Extensional Inequality and Complemented Subsets

The exposition rigorously establishes extensional properties, extensional subsets, and extensional inequality as central constructs for constructive mathematics. Complemented subsets are defined as pairs of strongly disjoint extensional subsets, with characteristic partial Boolean-valued functions. Operations (join, meet, complement) are defined via extensional properties, and complemented subsets form swap algebras (types $(I)$ or $(II)$ depending on operational choices). The algebraic structure is enriched by notions of inhabitedness, coinhabitedness, totality, and tightness.

Partial Boolean-Valued Functions and Constructive Gelfand Transform

Partial Boolean-valued functions are developed as a constructive counterpart to total Boolean homomorphisms. Equality and inequality of these functions are formulated extensionally, enabling the definition of swap characters as partial homomorphisms. The partial Gelfand transform is presented constructively, embedding a set into its second strong extensional (Boolean) partial dual, realized as a constructive embedding in the absence of PEM.

Swap Algebras and Separation

Swap algebras are formalized with explicit axioms matching complement and join/meet behavior. The paper analyzes the operational specifics for both type $(I)$ and $(II)$ swap algebras, highlighting that only type $CZF$0 enables representation theorems analogous to the classical Stone theorem. The characterization of separated swap algebras is grounded in the separation of points by swap characters (partial Boolean-valued homomorphisms) rather than maximal ideals.

Stone Representation Theorems

A constructive Stone representation theorem is proved for separated swap algebras of type $CZF$1, stating that a separated swap algebra embeds into the swap algebra of complemented subsets of its swap characters. The embedding is explicitly defined in terms of swap character evaluation, and the proof is carried out in intuitionistic logic; when restricted to swap algebras with Boolean inequality, the proof resides entirely in minimal logic.

For Boolean algebras, these results specialize to the constructive Stone representation theorem for separated Boolean algebras, with the embedding realized via total complemented subsets and total Boolean characters.

Stone–Čech Theorems and Reflective Subcategories

The authors prove a constructive Stone–Čech theorem for swap algebras of type $CZF$2, showing that every swap algebra corresponds to a separated swap algebra with equivalent swap character theory. This result parallels the classical topological Stone–Čech compactification and provides the categorical infrastructure of reflectors and adjoints. From the perspective of swap characters, working with separated algebras suffices.

Boolean Inequality and Minimal Logic

Sets with Boolean inequality are introduced to facilitate constructive proofs without resorting to Ex falso quodlibet (EFQ). Boolean inequality is defined on sets as an internal "falsum", enabling localized negation and constructive manipulation. The paper shows that many fundamental number systems (e.g., $CZF$3) are endowed with complete Boolean inequality, allowing the representation theorem proofs for corresponding swap algebras to operate within minimal logic.

Implications and Extensions

The constructive approach dissolves reliance on classical maximality (ideals) and choice axioms for Stone-type representation, recasting representation theory as a problem in equality and inequality. The implications are substantial for both constructive mathematical logic and classical theory: the results motivate reevaluation of classical algebraic and topological representation theorems from the constructive standpoint.

Potential future directions include:

• Extension of constructive representation theorems to commutative $CZF$4-algebras, employing partial characters and strong notion of separation.
• Development of topological Stone representation theorems for complemented topological spaces of type $CZF$5.
• Explore further generalizations to other algebraic structures (e.g., rings, modules) where swap algebraic machinery can be applied constructively.

Conclusion

The paper delivers a comprehensive constructive foundation for Stone representation theorems for separated swap algebras and Boolean algebras, including constructive Stone–Čech theorems and the introduction of sets with Boolean inequality. The results recast classical representation theorems in constructive terms, eliminate dependence on non-constructive principles, and establish categorical and algebraic tools for further research in constructive algebra and topology.