Transposition of variables is hard to axiomatize

Abstract

The function $p_{xy}$ that interchanges two logical variables $x,y$ in formulas is hard to describe in the following sense. Let F denote the Lindenbaum–Tarski formula-algebra of a finite-variable first-order logic, endowed with $p_{xy}$ as a unary function. We prove that each equational axiom system for the equational theory of F has to contain, for each finite n, an equation that contains together with $p_{xy}$ at least n algebraic variables, and each of the operations $\exists ,=,\vee$. This gives an answer to a problem raised by Johnson (1969) [30]: the class $RPE{A}_{\alpha }$ of representable polyadic equality algebras of a finite dimension $\alpha \ge 3$ cannot be axiomatized by adding finitely many equations to the equational theory of representable cylindric algebras of dimension α. Consequences for proof systems of finite-variable logic and for defining equations of polyadic equality algebras are given.

The proof uses a family of nonrepresentable polyadic equality algebras $A_n$ that are more and more nearly representable as n increases: their n-generated subalgebras as well as their proper reducts are representable. The lattice of subvarieties of $RPE{A}_{\alpha }$ is investigated and new open problems are asked about the interplay between the transposition operations and about generalizability of the results to infinite dimensions.