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Chrysafis Hartonas, Stone duality for lattice expansions, Logic Journal of the IGPL, Volume 26, Issue 5, October 2018, Pages 475–504, https://doi.org/10.1093/jigpal/jzy010
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Abstract
The Stone duality for bounded lattices by this author, with J.M. Dunn, is lifted in this article to a duality for lattices with operators. The dual frames of lattice expansions are two-sorted frames (X, |${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$|, Y), further equipped with an (n + 1)-ary relation |$\textit{R}_{\delta }$| and a dual relation |$\textit{R}^{\partial }_{\delta }$| for each n-ary lattice operator of some distribution type |$\delta $|. The closures of the generalized image operators generated by the relations are shown to be precisely the |$\sigma $|-extensions of the corresponding lattice operators and thus the full complex algebra of Galois-stable sets of the frame constitutes a concrete canonical extension of the lattice expansion. Thereby, the results presented in this article extend to the non-distributive case the classical Jónsson–Tarski results for Boolean algebras with operators and their extension to mere distributive lattices with operators. Consequently, the duality based approach to relational logic semantics is extended here to the case of logics dropping distribution. As an application example, we model the full BCK calculus. Both plain Kripke-type (two-sorted) frame semantics, as well as general (two-sorted) frame semantics are presented, the distinctive feature of the latter choice being that the interpretation of additional lattice operators (such as modal operators) is typically verbatim the same as in the distributive case, which is desirable in intended applications (such as temporal, or dynamic extensions of non-distributive lattice logic).
