Spectra of Arithmetic Frames through Priestley duality

Ponente(s): Sebastian David Melzer Rieger, Guram Bezhanishvili

Building on Priestley’s [5, 6] well-known duality for the category of bounded distributive lattices, Pultr and Sichler [7, 8] showed that Priestley duality restricts to a duality for the category of frames and frame homomorphisms. Recently we further restricted Pultr-Sichler duality to the categories of algebraic and arithmetic frames [1]. In this talk, we use this machinery to provide a new characterization of the d-nucleus on arithmetic frames, as introduced by Martinez and Zenk [4], and further studied by [2] and [3]. This approach allows us, among other things, to tackle some of the open problems stated in the literature. References [1] G. Bezhanishvili and S. Melzer. Algebraic frames in Priestley duality. arXiv:2306.06745, 2023. Submitted. [2] P. Bhattacharjee. Maximal d-elements of an algebraic frame. Order, 36(2):377–390, 2019. [3] T. Dube and L. Sithole. On the sublocale of an algebraic frame induced by the d-nucleus. Topology Appl., 263:90–106, 2019. [4] J. Martinez and E. R. Zenk. When an algebraic frame is regular. Algebra Universalis, 50(2):231–257, 2003. [5] H. A. Priestley. Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc., 2:186–190, 1970. [6] H. A. Priestley. Ordered topological spaces and the representation of distributive lattices. Proc. London Math. Soc., 24:507–530, 1972. [7] A. Pultr and J. Sichler. Frames in Priestley’s duality. Cahiers Topologie Géom. Différentielle Catég., 29(3):193–202, 1988. [8] A. Pultr and J. Sichler. A Priestley view of spatialization of frames. Cahiers Topologie Géom. Différentielle Catég., 41(3):225–238, 2000