Overlap Algebras: a Constructive Look at Complete Boolean Algebras
The notion of a complete Boolean algebra, although completely legitimate in constructive mathematics, fails to capture some natural structures such as the lattice of subsets of a given set. Sambin's notion of an overlap algebra, although classically equivalent to that of a complete Boolean alge...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Logical Methods in Computer Science e.V.
2020-02-01
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| Series: | Logical Methods in Computer Science |
| Subjects: | |
| Online Access: | https://lmcs.episciences.org/5417/pdf |
| _version_ | 1797268557221855232 |
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| author | Francesco Ciraulo Michele Contente |
| author_facet | Francesco Ciraulo Michele Contente |
| author_sort | Francesco Ciraulo |
| collection | DOAJ |
| description | The notion of a complete Boolean algebra, although completely legitimate in
constructive mathematics, fails to capture some natural structures such as the
lattice of subsets of a given set. Sambin's notion of an overlap algebra,
although classically equivalent to that of a complete Boolean algebra, has
powersets and other natural structures as instances. In this paper we study the
category of overlap algebras as an extension of the category of sets and
relations, and we establish some basic facts about mono-epi-isomorphisms and
(co)limits; here a morphism is a symmetrizable function (with classical logic
this is just a function which preserves joins). Then we specialize to the case
of morphisms which preserve also finite meets: classically, this is the usual
category of complete Boolean algebras. Finally, we connect overlap algebras
with locales, and their morphisms with open maps between locales, thus
obtaining constructive versions of some results about Boolean locales. |
| first_indexed | 2024-04-25T01:34:22Z |
| format | Article |
| id | doaj.art-509390250e0a498395eafdd405f32987 |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:34:22Z |
| publishDate | 2020-02-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-509390250e0a498395eafdd405f329872024-03-08T10:29:26ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742020-02-01Volume 16, Issue 110.23638/LMCS-16(1:13)20205417Overlap Algebras: a Constructive Look at Complete Boolean AlgebrasFrancesco CirauloMichele ContenteThe notion of a complete Boolean algebra, although completely legitimate in constructive mathematics, fails to capture some natural structures such as the lattice of subsets of a given set. Sambin's notion of an overlap algebra, although classically equivalent to that of a complete Boolean algebra, has powersets and other natural structures as instances. In this paper we study the category of overlap algebras as an extension of the category of sets and relations, and we establish some basic facts about mono-epi-isomorphisms and (co)limits; here a morphism is a symmetrizable function (with classical logic this is just a function which preserves joins). Then we specialize to the case of morphisms which preserve also finite meets: classically, this is the usual category of complete Boolean algebras. Finally, we connect overlap algebras with locales, and their morphisms with open maps between locales, thus obtaining constructive versions of some results about Boolean locales.https://lmcs.episciences.org/5417/pdfcomputer science - logic in computer sciencemathematics - logic |
| spellingShingle | Francesco Ciraulo Michele Contente Overlap Algebras: a Constructive Look at Complete Boolean Algebras Logical Methods in Computer Science computer science - logic in computer science mathematics - logic |
| title | Overlap Algebras: a Constructive Look at Complete Boolean Algebras |
| title_full | Overlap Algebras: a Constructive Look at Complete Boolean Algebras |
| title_fullStr | Overlap Algebras: a Constructive Look at Complete Boolean Algebras |
| title_full_unstemmed | Overlap Algebras: a Constructive Look at Complete Boolean Algebras |
| title_short | Overlap Algebras: a Constructive Look at Complete Boolean Algebras |
| title_sort | overlap algebras a constructive look at complete boolean algebras |
| topic | computer science - logic in computer science mathematics - logic |
| url | https://lmcs.episciences.org/5417/pdf |
| work_keys_str_mv | AT francescociraulo overlapalgebrasaconstructivelookatcompletebooleanalgebras AT michelecontente overlapalgebrasaconstructivelookatcompletebooleanalgebras |