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We offer here a particularly simple and direct construction of the congruence frame of a given frame, aka its assembly, from the bounded meet semilattice of differences of frame elements. The construction enables economical proofs of two of the assembly's known attributes, namely ultranormality and ultraparacompactness, as well as a proof that the assembly is an essential extension of the frame with the same essential closure as the frame. The major new result is that the assembly is free over its meet semilattice of differences. [ABSTRACT FROM AUTHOR]

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 this paper we first show that it is an essential extension of any L and that the maximal essential extensions of L and Sc(L) are isomorphic. The construction Sc is not functorial; this leads to the question of individual liftings of homomorphisms L → M to homomorphisms Sc(L) → Sc(M). This is trivial for Boolean L and easy for a wide class of spatial L, M. Then, we show that one can lift all h : L → 2 for weakly Hausdorff L (and hence the spectra of L and Sc(L) are naturally isomorphic), and finally present liftings of h : L → M for regular L and arbitrary Boolean M. [ABSTRACT FROM AUTHOR]

We use a landmark result in the theory of Riesz spaces – Freudenthal’s 1936 spectral theorem – to canonically represent any Archimedean lattice-ordered group G with a strong unit as a (non-separating) lattice-group of real-valued continuous functions on an appropriate G-indexed zero-dimensional compactification w G ⁢ Z G {w_{G}Z_{G}} of its space Z G {Z_{G}} of minimal prime ideals.The two further ingredients needed to establish this representation are the Yosida representation of G on its space X G {X_{G}} of maximal ideals, and the well-known continuous surjection of Z G {Z_{G}} onto X G {X_{G}}.We then establish our main result by showing that the inclusion-minimal extension of this representation of G that separates the points of Z G {Z_{G}} – namely, the sublattice subgroup of C ⁡ ( Z G ) {\operatorname{C}(Z_{G})} generated by the image of G along with all characteristic functions of clopen (closed and open) subsets of Z G {Z_{G}} which are determined by elements of G – is precisely the classical projectable hull of G.Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit. [ABSTRACT FROM AUTHOR]

We address what might be termed the reverse reflection problem: given a monoreflection from a category A onto a subcategory B, when is a given objectB∈ B the reflection of a proper subobject? We start with a well known specific instance of this problem, namely the fact that a compact metric space is never the Čech-Stone compactification of a proper subspace. We show that this holds also in the pointfree setting, i.e., that a compact metrizable locale is never the Čech-Stone compactification of a proper sublocale. This is a stronger result than the classical one, but not because of an increase in scope; after all, assuming weak choice prin­ciples, every compact regular locale is the topology of a compact Hausdorff space. The increased strength derives from the conclusion, for in general a space has many more sublocales than subspaces. We then extend the analysis from metric locales to the broader class of perfectly normal locales, i.e., those whose frame of open sets consists entirely of cozero elements. We include a second proof of these results which is purely algebraic in character. At the opposite extreme from these results, we show that an extremally disconnected locale is a compactification of each of its dense sublocales. Finally, we analyze the same phenomena, also in the pointfree setting, for the 0-dimensional compact reflec­tion and for the Lindelöf reflection. [ABSTRACT FROM PUBLISHER]

It is well known that a locale is subfit i¤ each of its open sublocales is a join of closed ones, and fit i¤ 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 i¤ each of its sublocales is a meet of closed ones, but it is not the case that a locale is subfit i¤ each of its sublocales is a join of closed ones. Thus we are led to take up the very natural question of which locales have the feature that every sublocale is a join of closed sublocales. In this note we show that these are precisely the subfit locales which are scattered in the point-free sense of [13], and we add a variation for spatial frames. [ABSTRACT FROM AUTHOR]