Details
| Original language | English |
|---|---|
| Article number | 13 |
| Number of pages | 23 |
| Journal | Algebra universalis |
| Volume | 83 |
| Issue number | 2 |
| Early online date | 19 Mar 2022 |
| Publication status | Published - May 2022 |
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.
Keywords
- Adjoint map, Complement, Implicative semilattice, Localic map, Nuclear range, Sublocale
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
- Mathematics(all)
- Logic
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In: Algebra universalis, Vol. 83, No. 2, 13, 05.2022.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Adjoint maps between implicative semilattices and continuity of localic maps
AU - Erné, Marcel
AU - Picado, Jorge
AU - Pultr, Aleš
N1 - Funding Information: J. Picado gratefully acknowledges financial support from the Centre for Mathematics of the University of Coimbra (UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES). The research of A. Pultr was supported by the Department of Applied Mathematics (KAM) of Charles University (Prague)
PY - 2022/5
Y1 - 2022/5
N2 - 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.
AB - 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.
KW - Adjoint map
KW - Complement
KW - Implicative semilattice
KW - Localic map
KW - Nuclear range
KW - Sublocale
UR - http://www.scopus.com/inward/record.url?scp=85126816202&partnerID=8YFLogxK
U2 - 10.1007/s00012-022-00767-4
DO - 10.1007/s00012-022-00767-4
M3 - Article
AN - SCOPUS:85126816202
VL - 83
JO - Algebra universalis
JF - Algebra universalis
SN - 0002-5240
IS - 2
M1 - 13
ER -