Publications

Authors:

Misselbeck-Wessel, Daniel; Petrakis, Iosif

Title:

Complemented subsets and Boolean-valued, partial functions

Year:

2024

Type of item:

Articolo in Rivista

Tipologia ANVUR:

Articolo su rivista

Language:

Inglese

Format:

Elettronico

Referee:

Sì

Name of journal:

COMPUTABILITY

ISSN of journal:

2211-3576

Page numbers:

1-33

Keyword:

Boolean algebras, Boolean rings, swap algebras, swap rings

Short description of contents:

We study the two algebras of complemented subsets that were introduced in the constructive development of the Daniell approach to measure and integration within Bishop-style constructive mathematics. We present their main properties both for the so-called here categorical complemented subsets and for the extensional complemented subsets. We translate constructively the classical bijection between subsets and Boolean-valued, total functions by establishing a bijection between complemented subsets (categorical or extensional) and Boolean-valued, partial functions (categorical or extensional). The role of Myhill's axiom of non-choice in the equivalence between categorical and extensional subsets is discussed. We introduce swap algebras of type (I) and (II) as an abstract version of Bishop’s algebras of complemented subsets of type (I) and (II), respectively, and swap rings as an abstract version of the structure of Boolean-valued partial functions on a set. Our examples of swap algebras and swap rings together with the included here results indicate that their theory is a certain generalisation of the theory of Boolean algebras and Boolean rings, a fact which we find interesting both from a constructive and a classical point of view.

Web page:

https://content.iospress.com/journals/computability/Pre-press/Pre-press

Product ID:

138011

Handle IRIS:

11562/1120029

Last Modified:

April 4, 2025

Bibliographic citation:

Misselbeck-Wessel, Daniel; Petrakis, Iosif, Complemented subsets and Boolean-valued, partial functions «COMPUTABILITY» , 2024 , pp. 1-33

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