Nuclear ranges in implicative semilattices

authored by

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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Algebra universalis

Volume

83

No. of pages

22

ISSN

0002-5240

Publication date

05.2022

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Algebra and Number Theory, Logic

Electronic version(s)

https://doi.org/10.1007/s00012-022-00768-3 (Access: Open)