Erdős Graphs Resolve Fine's Canonicity Problem

We show that there exist 2ℵ0 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 (a monomodal example can then be obtained using simulation results of Thomason). The constructions use the result of Erdos that there are finite graphs with arbitrarily large chromatic number and girth.

Atomless varieties

We define a nontrivial variety of boolean algebras with operators such that every member of the variety is atomless. This shows that not every variety of boolean algebras with operators is generated by its atomic members, and thus establishes a strong incompleteness result in (multi-)modal logic.

Amalgamation in relation algebras

We investigate amalgamation properties of relational type algebras. Besides purely algebraic interest, amalgamation in a class of algebras is important because it leads to interpolation results for the logic corresponding to that class (cf. [15]). The multi-modal logic corresponding to relational type algebras became known under the name of “arrow logic” (cf. [18, 17]), and has been studied rather extensively lately (cf. [10]). Our research was inspired by the following result of Andréka et al. [1].Let K be a class of relational type algebras such that(i) composition is associative,(ii) K is a class of boolean algebras with operators, and(iii) K contains the representable relation algebras RRA.Then the equational theory of K is undecidable.On the other hand, there are several classes of relational type algebras (e.g., NA, WA denned below) whose equational (even universal) theories are decidable (cf. [13]). Composition is not associative in these classes. Theorem 5 indicates that also with respect to amalgamation (a very weak form of) associativity forms a borderline. We first recall the relevant definitions.