Publications
McKinsey-Tarski algebras and Raney extensions Preprint
Guram Bezhanishvili, Ranjitha Raviprakash, Anna Laura Suarez, and Joanne Walters-Wayland
arXiv preprint (2025). PDF links: arXiv
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Abstract. We introduce the notion of Raney morphism between MT-algebras and show that the resulting category is equivalent to the category of Raney extensions. This is done by generalizing the construction of the Funayama envelope of a frame. The resulting notion of the \(T_D\)-hull of a Raney extension generalizes that of the \(T_D\)-hull of a frame.
Local compactness does not always imply spatiality To appear
Guram Bezhanishvili, Sebastian Melzer, Ranjitha Raviprakash, and Anna Laura Suarez
Questions and Answers in General Topology (2025). PDF links: arXiv
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Abstract. It is a well-known result in pointfree topology that every locally compact frame is spatial. Whether this result extends to MT-algebras (McKinsey–Tarski algebras) was an open problem. We resolve it in the negative by constructing a locally compact sober MT-algebra which is not spatial. We also revisit Nöbeling’s largely overlooked approach to pointfree topology from the 1950s. We show that his separation axioms are closely related to those in the theory of MT-algebras with the notable exception of Hausdorffness. We prove that Nöbeling’s Spatiality Theorem implies the well-known Isbell Spatiality Theorem. We then generalize Nöbeling’s Spatiality Theorem by proving that each locally compact \(T_{1/2}\)-algebra is spatial. The proof utilizes the fact that every non-trivial \(T_{1/2}\)-algebra contains a closed atom, which we show is equivalent to the axiom of choice.
The Funayama envelope as the TD-hull of a frame
Guram Bezhanishvili, Ranjitha Raviprakash, Anna Laura Suarez, and Joanne Walters-Wayland
Theory and Applications of Categories (2025). PDF links: journal arXiv
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Abstract. We introduce proximity morphisms between MT-algebras and show that the resulting category is equivalent to the category of frames. This is done by utilizing the Funayama envelope of a frame, which is viewed as the \(T_D\)-hull. Our results have some spatial ramifications, including a generalization of the \(T_D\)-duality of Banaschewski and Pultr.
Local Compactness in MT-algebras
Guram Bezhanishvili and Ranjitha Raviprakash
Topology Proceedings (2025). PDF links: journal arXiv
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Abstract. In our previous work, we introduced McKinsey–Tarski algebras (MT-algebras for short) as an alternative pointfree approach to topology. Here, we study local compactness in MT-algebras. We establish the Hofmann–Mislove theorem for sober MT-algebras, which we use to develop the MT-algebra versions of such well-known dualities in pointfree topology as Hofmann–Lawson, Isbell, and Stone dualities. This yields a new perspective on these classic results.
McKinsey-Tarski algebras: An alternative pointfree approach to topology
Guram Bezhanishvili and Ranjitha Raviprakash
Topology Applications (2023). PDF links: doi arXiv
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Abstract. McKinsey and Tarski initiated the study of interior algebras. We propose complete interior algebras as an alternative pointfree approach to topology. We term these algebras McKinsey–Tarski algebras or simply MT-algebras. Associating with each MT-algebra the lattice of its open elements defines a functor from the category of MT-algebras to the category of frames, which we study in depth. We also study the dual adjunction between the categories of MT-algebras and topological spaces, and show that MT-algebras provide a faithful generalization of topological spaces. Our main emphasis is on developing a unified approach to separation axioms in the language of MT-algebras, which generalizes separation axioms for both topological spaces and frames.