Adjoint maps between implicative semilattices and continuity of localic maps

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. © 2022, The Author(s)

Nuclear ranges in implicative semilattices

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. © 2022, The Author(s)