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Title: "Uniform Insertion Theorem for Frames vs. Blair and Lane’s General Insertion Theorem"
Speaker : Ana Belen Avilez Garcia (Chapman University, USA)
Abstract:In [1] the authors generalized Preiss and Vilimovsky's insertion theorem for uniform frames, and in [2] the authors proved the general insertion theorem of Blair and Lane in the point-free setting. In this talk we will compare both results.
We will first give the necessary background to understand both results. In particular, we will introduce the relation of farness for elements and sublocales of a locale. We will define uniform continuity for general, not necessarily continuous, real-valued functions, and then show that uniform continuity can be characterized by the notion of farness. We will then present the insertion theorem for uniform frames.
Finally, we will compare both insertion theorems highlighting the similarities between the two results and stressing the importance the two relations in sublocales (namely, complete separation and farness) play in these insertion-type results.
References:
[1] Arrieta, I. and Avilez, A. B. (2023). A general insertion theorem for uniform locales, J. Pure Appl. Algebra, 227(7), 107320.
[2] J. Gutiérrez García and T. Kubiak, General insertion and extension theorems for localic real functions, J. Pure Appl. Algebra, 215 (2011) 1198-1204.
Title: "Priestley duality for frames"
Speaker : Sebastian Melzer (New Mexico State University, USA)
Abstract: We explore the duality between frames and L-spaces, refining Priestley duality for bounded distributive lattices. We review key properties of Priestley spaces of frames, alongside a topological characterization of frame homomorphisms. Within this framework, we examine (Scott) open filters and nuclei, providing an analogue of the Hofmann-Mislove theorem, and obtain a new proof that every open filter is admissible.
Title: "McKinsey-Tarski algebras" Seminar slides
Speaker : Ranjitha Raviprakash (New Mexico State University, USA)
Abstract: This talk surveys McKinsey-Tarski (MT) algebras. We examine the category of MT-algebras and its connections to the categories of topological spaces and frames, highlighting how classical notions from topology and frame theory, such as separation axioms, compactness, and local compactness, are generalized and unified within this framework. Additionally, we discuss adaptations of the Hofmann-Mislove theorem and well-known dualities, including those of Hofmann-Lawson, Isbell, and Stone. The talk includes open problems in the area, inviting further investigation.
Title: "A pointfree theory of T_0 spaces"
Speaker : Anna Laura Suarez (University of Coimbra, PT)
Abstract: In classical pointfree topology, one regards frames as abstract topological spaces, in virtue of an adjunction between the two categories. The fixpoints on the Top side are the sober spaces. In an alternative approach, known as T_D duality, the spectrum functor is modified so that the fixpoints are the T_D spaces. None of the two dualities subsumes the other. We present a duality that extends both, such that the fixpoints in Top are all T_0 spaces. The main idea is that we identify a space X with the embedding O(X)-->Sat(X) of the opens into the saturated sets. Our pointfree category is that of Raney extensions of frames, frames embedded into coframes satisfying certain properties. By embedding both the sober and the T_D duality into the same large picture, we confirm the idea of the sober spectrum and the T_D spectrum of a frame being, respectively, the largest and the smallest T_0 space associated with a frame L: we show that every frame admits the largest and the smallest Raney extension over it, and that the spectra of these are, respectively, the classical spectrum of L and the T_D spectrum. We characterize both axioms algebraically, in terms of the lattice of saturated sets, thus obtaining natural definitions of sober and T_D Raney extensions. We show that every Raney extension admits a sobrification (i.e. a sober coreflection), and that under a suitable restriction of morphisms it also admits a T_D reflection.
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