Finite \(H\)-dimension does not imply expressive completeness
The monadic first-order language (formulas in this language are called monadic) is obtained from a given first-order language by adding a set of monadic predicates. A monadic first-order formula with a single free variable is called connective. A class of first-order structures \(K\) has \(H\)-dimension \(d\) if \(d\) is the least number such that every monadic formula is \(K\)-equivalent to a monadic formula using at most \(d\) bound variables. It is known that if a class of structures has a finite basis of connectives, then it has finite \(H\)-dimension. In this paper the other direction (a conjecture of D. Gabbay) is refuted.
scientific article; zbMATH DE number 683099