Abstract
AbstractA nucleus on a meet-semilattice A is a closure operation that preserves binary meets. The nuclei form a semilattice $$\mathrm{N }A$$
N
A
that is isomorphic to the system $${\mathcal {N}}A$$
N
A
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.
Funder
Gottfried Wilhelm Leibniz Universität Hannover
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
Reference45 articles.
1. Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. The Joy of Cats. Dover Publ. Inc., Dover (2009)
2. Baer, R.: On closure operators. Arch. Math. 10, 261–266 (1959)
3. Banaschewski, B.: Another look at the localic Tychonoff theorem. Comment. Math. Univ. Carolin. 29, 647–656 (1988)
4. Beazer, R., Macnab, D.S.: Modal operators on Heyting algebras. Colloquium Math. XVI, 1–12 (1979)
5. Bergmann, G.: Multiplicative closures. Port. Math. 11, 169–172 (1952)