# Ilya Shapirovsky: Locally finite varieties of modal algebras

Place: Fields Institute (Room 210)
Date: June 15, 2018 (13:30-15:00)
Speaker: Ilya Shapirovsky
Title: Locally finite varieties of modal algebras
Abstract: A modal algebra is a Boolean algebra enriched with an additive operation. Equational theories of modal algebras are called modal logics. A logic L is said to be n-tabular if, up to the equivalence in L, there exist only finitely many n-variable formulas.  L is locally tabular if it is n-tabular for all finite n. Algebraically, n-tabularity of a logic means that its n-generated free algebra is finite (thus,  local tabularity of a logic is equivalent to local finiteness of its variety).

It is known that a variety of closure algebras  is locally finite iff its one-generated free algebra is finite (Larisa Maksimova, 1975). The following question has been open since 1970s: does this equivalence hold for every variety of modal algebras? (The analogous problem is open for varieties of Heyting algebras: does 2-tabularity of an intermediate logic imply local tabularity?)

Recently, in our joint work with Valentin Shehtman, it was shown how local tabularity of modal logics can be characterized in terms of partitions of relational structures. I will discuss this criterion and then use it to construct the first example  of a 1-tabular but not locally tabular modal logic.

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