On joins of complemented sublocales

Igor Arrieta Torres

doi:10.1007/s00012-021-00757-y

On joins of complemented sublocales

Igor Arrieta Torres^*

^*Corresponding author for this work

Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

The system S_c(L) consisting of joins of closed sublocales of a locale L is known to be a frame, and for L subfit it coincides with the Booleanization S_b(L) of the coframe of sublocales of L. In this paper, we study S_b(L) for a general locale L. We show that S_c(L) is always a subframe of S_b(L) . Moreover, if X is a T_D -space, we prove that S_b(Ω(X)) is precisely the set of classical subspaces of X, and that a locale L is T_D -spatial iff the Boolean algebra S_b(L) is atomic. Some functoriality properties of S_b(L) are also studied.

Original language	English
Article number	1
Number of pages	11
Journal	Algebra Universalis
Volume	83
Early online date	13 Nov 2021
DOIs	https://doi.org/10.1007/s00012-021-00757-y
Publication status	Published - Feb 2022

Keywords

Locale
Frame
Sublocale
Booleanization
Induced sublocale
Complemented sublocale
Subfit locale
TD-axiom

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10.1007/s00012-021-00757-y

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abstract = "The system Sc(L) consisting of joins of closed sublocales of a locale L is known to be a frame, and for L subfit it coincides with the Booleanization Sb(L) of the coframe of sublocales of L. In this paper, we study Sb(L) for a general locale L. We show that Sc(L) is always a subframe of Sb(L) . Moreover, if X is a TD -space, we prove that Sb(Ω(X)) is precisely the set of classical subspaces of X, and that a locale L is TD -spatial iff the Boolean algebra Sb(L) is atomic. Some functoriality properties of Sb(L) are also studied. ",

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N2 - The system Sc(L) consisting of joins of closed sublocales of a locale L is known to be a frame, and for L subfit it coincides with the Booleanization Sb(L) of the coframe of sublocales of L. In this paper, we study Sb(L) for a general locale L. We show that Sc(L) is always a subframe of Sb(L) . Moreover, if X is a TD -space, we prove that Sb(Ω(X)) is precisely the set of classical subspaces of X, and that a locale L is TD -spatial iff the Boolean algebra Sb(L) is atomic. Some functoriality properties of Sb(L) are also studied.

AB - The system Sc(L) consisting of joins of closed sublocales of a locale L is known to be a frame, and for L subfit it coincides with the Booleanization Sb(L) of the coframe of sublocales of L. In this paper, we study Sb(L) for a general locale L. We show that Sc(L) is always a subframe of Sb(L) . Moreover, if X is a TD -space, we prove that Sb(Ω(X)) is precisely the set of classical subspaces of X, and that a locale L is TD -spatial iff the Boolean algebra Sb(L) is atomic. Some functoriality properties of Sb(L) are also studied.

KW - Locale

KW - Frame

KW - Sublocale

KW - Booleanization

KW - Induced sublocale

KW - Complemented sublocale

KW - Subfit locale

KW - TD-axiom

U2 - 10.1007/s00012-021-00757-y

DO - 10.1007/s00012-021-00757-y

M3 - Article

SN - 0002-5240

VL - 83

JO - Algebra Universalis

JF - Algebra Universalis

M1 - 1

ER -

On joins of complemented sublocales

Abstract

Keywords

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