# Generating sublocales by subsets and relations: a tangle of adjunctions

Generating sublocales by subsets and relations: a tangle of adjunctions Generalizing the obvious representation of a subspace \$\${Y \subseteq X}\$\$ Y ⊆ X as a sublocale in Ω(X) by the congruence \$\${\{(U, V ) | U\cap Y = V \cap Y\}}\$\$ { ( U , V ) | U ∩ Y = V ∩ Y } , one obtains the congruence \$\${\{(a, b) |\mathfrak{o}(a) \cap S = \mathfrak{o}(b) \cap S\}}\$\$ { ( a , b ) | o ( a ) ∩ S = o ( b ) ∩ S } , first with sublocales S of a frame L, which (as it is well known) produces back the sublocale S itself, and then with general subsets \$\${S\subseteq L}\$\$ S ⊆ L . The relation of such S with the sublocale produced is studied (the result is not always the sublocale generated by S). Further, we discuss in general the associated adjunctions, in particular that between relations on L and subsets of L and view the aforementioned phenomena in this perspective. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png algebra universalis Springer Journals

# Generating sublocales by subsets and relations: a tangle of adjunctions

, Volume 78 (1) – May 22, 2017
14 pages

Publisher
Springer International Publishing
Subject
Mathematics; Algebra
ISSN
0002-5240
eISSN
1420-8911
D.O.I.
10.1007/s00012-017-0446-z
Publisher site
See Article on Publisher Site

### Abstract

Generalizing the obvious representation of a subspace \$\${Y \subseteq X}\$\$ Y ⊆ X as a sublocale in Ω(X) by the congruence \$\${\{(U, V ) | U\cap Y = V \cap Y\}}\$\$ { ( U , V ) | U ∩ Y = V ∩ Y } , one obtains the congruence \$\${\{(a, b) |\mathfrak{o}(a) \cap S = \mathfrak{o}(b) \cap S\}}\$\$ { ( a , b ) | o ( a ) ∩ S = o ( b ) ∩ S } , first with sublocales S of a frame L, which (as it is well known) produces back the sublocale S itself, and then with general subsets \$\${S\subseteq L}\$\$ S ⊆ L . The relation of such S with the sublocale produced is studied (the result is not always the sublocale generated by S). Further, we discuss in general the associated adjunctions, in particular that between relations on L and subsets of L and view the aforementioned phenomena in this perspective.

### Journal

algebra universalisSpringer Journals

Published: May 22, 2017

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