Adjoint maps between implicative semilattices and continuity of localic maps

authored by

Marcel Erné, Jorge Picado, Aleš Pultr

Abstract

We study residuated homomorphisms (r-morphisms) and their adjoints, the so-called localizations (or l-morphisms), between implicative semilattices, because these objects may be characterized as semilattices whose unary meet operations have adjoints. Since left resp. right adjoint maps are the residuated resp. residual maps (having the property that preimages of principal downsets resp. upsets are again such), one may not only regard the l-morphisms as abstract continuous maps in a pointfree framework (as familiar in the complete case), but also characterize them by concrete closure-theoretical continuity properties. These concepts apply to locales (frames, complete Heyting lattices) and provide generalizations of continuous and open maps between spaces to an algebraic (not necessarily complete) pointfree setting.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

Charles University
University of Coimbra

Type

Article

Journal

Algebra universalis

Volume

83

No. of pages

23

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/s00012-022-00767-4 (Access: Open)