A dual characterisation of simple and subdirectly-irreducible temporal Heyting algebras

Mathematics > Logic [Submitted on 12 May 2025 (v1), last revised 16 Jun 2026 (this version, v2)] Title:A dual characterisation of simple and subdirectly-irreducible temporal Heyting algebras View PDFAbstract:We establish an Esakia duality for the categories of temporal Heyting algebras and temporal Esakia spaces. This includes a proof of contravariant equivalence and a congruence/filter/closed-upset correspondence. We then study two notions of « reachability » on the relevant spaces/frames and show their equivalence in the finite case. We use these notions of reachability to give both lattice-theoretic and dual order-topological characterisations of simple and subdirectly-irreducible temporal Heyting algebras. Finally, we apply our duality results to prove the relational and algebraic finite model property for the temporal Heyting calculus. This, in conjunction with the proven characterisations, allows us to prove a relational completeness result that combines finiteness and the frame property dual to subdirect-irreducibility, giving us a class of finite, well-understood frames for the logic. Submission history From: David Quinn Alvarez [view email][v1] Mon, 12 May 2025 19:24:47 UTC (33 KB) [v2] Tue, 16 Jun 2026 08:15:18 UTC (35 KB) Current browse context: math.LO References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.