Abstract

We show that there exist 2^ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames. The constructions use the result of Erd os that there are finite graphs with arbitrarily large chromatic number and girth.

Citations