Algebras of Complemented Subsets

Complemented subsets were introduced by Bishop, in order to avoid complementation in terms of negation. In his two approaches to measure theory Bishop used two sets of operations on complemented subsets. Here we study these two algebras and we introduce the notion of Bishop algebra as an abstraction...

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Bibliographic Details
Published inRevolutions and Revelations in Computability Vol. 13359; pp. 246 - 258
Main Authors Petrakis, Iosif, Wessel, Daniel
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 2022
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Bishop sets
complemented subsets
partial functions
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ISBN3031087399
9783031087394
ISSN0302-9743
1611-3349
DOI10.1007/978-3-031-08740-0_21

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Summary:Complemented subsets were introduced by Bishop, in order to avoid complementation in terms of negation. In his two approaches to measure theory Bishop used two sets of operations on complemented subsets. Here we study these two algebras and we introduce the notion of Bishop algebra as an abstraction of their common structure. We translate constructively the classical bijection between subsets and boolean-valued functions by establishing a bijection between the proper classes of complemented subsets and of strongly extensional, boolean-valued, partial functions. Avoiding negatively defined concepts, most of our results are within minimal logic.
ISBN:3031087399
9783031087394
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-031-08740-0_21