Details
| Original language | English |
|---|---|
| Article number | 18 |
| Number of pages | 22 |
| Journal | Algebra universalis |
| Volume | 83 |
| Issue number | 2 |
| Early online date | 12 Apr 2022 |
| Publication status | Published - May 2022 |
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.
Keywords
- Closure operation, Closure range, Heyting lattice, Implicative semilattice, l-ideal, Nuclear range, Nucleus
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
- Mathematics(all)
- Logic
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In: Algebra universalis, Vol. 83, No. 2, 18, 05.2022.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Nuclear ranges in implicative semilattices
AU - Erné, Marcel
N1 - Funding information: Open Access funding enabled and organized by Projekt DEAL.
PY - 2022/5
Y1 - 2022/5
N2 - 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.
AB - 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.
KW - Closure operation
KW - Closure range
KW - Heyting lattice
KW - Implicative semilattice
KW - l-ideal
KW - Nuclear range
KW - Nucleus
UR - http://www.scopus.com/inward/record.url?scp=85128191198&partnerID=8YFLogxK
U2 - 10.1007/s00012-022-00768-3
DO - 10.1007/s00012-022-00768-3
M3 - Article
AN - SCOPUS:85128191198
VL - 83
JO - Algebra universalis
JF - Algebra universalis
SN - 0002-5240
IS - 2
M1 - 18
ER -