# On an aspect of scatteredness in the point-free setting

@article{Ball2016OnAA, title={On an aspect of scatteredness in the point-free setting}, author={R. Ball and J. Picado and A. Pultr}, journal={Portugaliae Mathematica}, year={2016}, volume={73}, pages={139-152} }

It is well known that a locale is subfit iff each of its open sublocales is a join of closed ones, and fit iff each of its closed sublocales is a meet of open ones. This formulation, however, exaggerates the parallelism between the behavior of fitness and subfitness. For it can be shown that a locale is fit iff each of its sublocales is a meet of closed ones, but it is not the case that a locale is subfit iff each of its sublocales is a join of closed ones. Thus we are led to take up the very… Expand

#### 9 Citations

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Sublocales that are joins of closed ones constitute a frame Sc(L) embedded as a join-sublattice into the coframe S(L) of sublocales of L. We prove that in the case of subfit L it is a subcolocale of… Expand

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Abstract Point-free modeling of mappings that are not necessarily continuous has been so far based on the extension of a frame to its frame of sublocales, mimicking the replacement of a topological… Expand

Exact Filters and Joins of Closed Sublocales

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We prove, for a general frame, that the sublocales that can be represented as joins of closed ones are, somewhat surprisingly, in a natural one-to-one correspondence with the filters closed under… Expand

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Abstract Arhangel'skii has defined a topological space X to be densely normal if it has a dense subspace Y such that any two disjoint closed subsets of X that are closures in X of some closed subsets… Expand

New Aspects of Subfitness in Frames and Spaces

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Another necessary and sufficient condition for subfitness presented is the validity of the meet formula for the Heyting operation, which was so far known only under much stronger conditions. Expand

Some aspects of (non) functoriality of natural discrete covers of locales

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Abstract The frame Sc(L) generated by closed sublocales of a locale L is known to be a natural Boolean (“discrete”) extension of a subfit L; also it is known to be its maximal essential extension. In… Expand

Exact and Strongly Exact Filters

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A characteristic of the coframe of meets of open sublocales as the dual to the frame of strongly exact filters is presented. Expand

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We discuss the congruences θ that are connected as elements of the (totally disconnected) congruence frame CL, and show that they are in a one-to-one correspondence with the completely prime elements… Expand

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We show that every frame can be essentially embedded in a Boolean frame, and that this embedding is the maximal essential extension of the frame in the sense that it factors uniquely through any… Expand

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