Nuclear ranges in implicative semilattices

verfasst von

Marcel Erné

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.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Typ

Artikel

Journal

Algebra universalis

Band

83

Anzahl der Seiten

22

ISSN

0002-5240

Publikationsdatum

05.2022

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Algebra und Zahlentheorie, Logik

Elektronische Version(en)

https://doi.org/10.1007/s00012-022-00768-3 (Zugang: Offen)