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arXiv · 2606.08176

Constructive Stone representations for separated swap and Boolean algebras

Daniel Misselbeck-Wessel·Iosif Petrakis
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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.

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Daniel Misselbeck-Wessel, Iosif Petrakis. 2026-06-06. Constructive Stone representations for separated swap and Boolean algebras. https://arxiv.org/abs/2606.08176

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