Induced congruences in \(\sigma\)-frames

Given a \( \sigma \)-space \( X \) and its lattice of open sets \( \mathcal{O}(X) \), we will study congruences on the \( \sigma \)-frame \( \mathcal{O}(X) \) representing the subspaces of \( X \) (referred to as induced congruences). We will show that when \( X \) is a \( T_D \) \( \sigma \)-space, there is a bijection between the subspaces of \( X \) and the congruences induced by them.

While in the setting of frames and locales the sublocales of a given locale \( L \) have a concrete description as subsets of \( L \) (since frame homomorphisms have right adjoints), similar descriptions do not hold in the setting of \( \sigma \)-frames and \( \sigma \)-locales, which presents a challenge when we try to represent subspaces of a \( \sigma \)-space as congruences in \( \sigma \)-frames.

Nonetheless, through the (contravariant) adjunction between the category of \( \sigma \)-spaces and \( \sigma \)-continuous maps and the category of \( \sigma \)-frames and \( \sigma \)-frame homomorphisms, we can define and describe a congruence induced by a subspace \( Y\subseteq X \). This correspondence yields a map \( \pi:\mathcal{P}(X) \to\mathcal{C}(\mathcal{O}(X)) \) from the powerset of \( X \) to the congruence lattice of \( \mathcal{O}(X) \), which is injective when \( X \) is a \( T_D \) \( \sigma \)-space. And when the axiom \( T_D \) is satisfied, we have \( \pi(\mathcal P(X))\subseteq\mathcal C_b(\mathcal{O}(X)) \), where \( \mathcal C_b(\mathcal{O}(X)) \) is the subset of \( \mathcal C(\mathcal{O}(X)) \) consisting of meets of complemented congruences.

Date:  2023-12-05
Start Time:   15:00
Speaker:  Raquel Bernardes (CMUC, Univ. Coimbra)
Institution:  CMUC, Univ. Coimbra
Place:  Sala 5.5, DMUC
Research Groups: -Algebra, Logic and Topology
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