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Generalizing the obvious representation of a subspace Y⊆X as a sublocale in Ω(X) by the congruence {(U,V)|U∩Y=V∩Y}, one obtains the congruence {(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⊆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.


This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Algebra Universalis, volume 78, issue 1, in 2017 following peer review. The final publication may differ and is available at Springer via DOI: 10.1007/s00012-017-0446-z.

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