Zhiguang ZhaoTaishan University · School of Mathematics and Statistics
Exploring relational, topological and algebraic aspects of correspondence theory
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For Boolean algebras with operators, we know that if a variety/universal class of BAOs is closed under taking MacNeille completions, then it is also closed under taking canonical extensions. Universal classes correspond to algebra classes defined by formulas universally quantified, so they are in the Pi_1 hierarchy. If we consider Pi_2 statements (i.e. statements of the form forall......exists......) and their corresponding inductive classes (closed under taking union of chains or directed limits), would the situation be the same or different?
This part of research concerns correspondence theory which is more convenient to do with relational semantics, as a supplement of the unified correspondence project which is more on the algebraic side.
This is a line of research started in 2009 and aimed at understanding the phenomenon of Sahlqvist correspondence and canonicity from an algebraic viewpoint. Thanks to Stone-type dualities, we have been able to reformulate the Sahlqvist mechanism in terms of the order-theoretic properties of the algebraic interpretation of the logical connectives. In its turn, this order-theoretic reformulation has made it possible to extend the state-of-the-art in Sahlqvist theory from classical normal modal logic to classes of logics algebraically captured by normal and regular (distributive) lattice expansions, mu calculi, hybrid logics, and many valued logics.