Nuclear ranges in implicative semilattices

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Autoren

  • Marcel Erné
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Details

OriginalspracheEnglisch
Aufsatznummer18
Seitenumfang22
FachzeitschriftAlgebra universalis
Jahrgang83
Ausgabenummer2
Frühes Online-Datum12 Apr. 2022
PublikationsstatusVeröffentlicht - Mai 2022

Abstract

A nucleus on a meet-semilattice A is a closure operation that preserves binary meets. The nuclei form a semilattice N A that is isomorphic to the system NA of all nuclear ranges, ordered by dual inclusion. The nuclear ranges are those closure ranges which are total subalgebras (l-ideals). Nuclei have been studied intensively in the case of complete Heyting algebras. We extend, as far as possible, results on nuclei and their ranges to the non-complete setting of implicative semilattices (whose unary meet translations have adjoints). A central tool are so-called r-morphisms, that is, residuated semilattice homomorphisms, and their adjoints, the l-morphisms. Such morphisms transport nuclear ranges and preserve implicativity. Certain completeness properties are necessary and sufficient for the existence of a least nucleus above a prenucleus or of a greatest nucleus below a weak nucleus. As in pointfree topology, of great importance for structural investigations are three specific kinds of l-ideals, called basic open, boolean and basic closed.

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Nuclear ranges in implicative semilattices. / Erné, Marcel.
in: Algebra universalis, Jahrgang 83, Nr. 2, 18, 05.2022.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Erné M. Nuclear ranges in implicative semilattices. Algebra universalis. 2022 Mai;83(2):18. Epub 2022 Apr 12. doi: 10.1007/s00012-022-00768-3
Erné, Marcel. / Nuclear ranges in implicative semilattices. in: Algebra universalis. 2022 ; Jahrgang 83, Nr. 2.
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