Three interpolation theorems for typeless logics

We prove three interpolation theorems for typless logics with equality. Typless logics, introduced by Henkin and Tarski, are algebraizable extensions of first-order logic allowing infinitary predicate symbols. Δα (α an infinite ordinal) denotes the language of a typless logic with α many variables. As a sample, we shall prove:THEOREM 1Let Δα be a language (possibly uncountable). Let ψ, ø, be formulas such that $${\models \phi \rightarrow \psi}$$ Then there exist m<α, iε^m α and ŋ a formula containing only atomic formulas occurring in ψ and ø, in short such that $${\models \forall v_{i_0} \ldots \models \forall v_{i_{m-1}} \phi \rightarrow \eta \ and \ \models \eta \rightarrow \exists v_{i_0} \ldots v_{i_{m-1}} \psi}$$ Fmr denotes the set of all formulas in a language. Now assuming, that everything is countable, we show that there exists two unary connectives, such that if Fm denotes the formulas in this expanded language, thenTHEOREM 2 $${(\forall \psi, \phi \in Fm_r)[\models \phi \rightarrow \psi \Rightarrow \exists \eta \in Fm, atf(\eta) \subseteq atf (\phi) \cap atf (\psi) \models \phi \rightarrow \eta \ and \ \models \eta \rightarrow \psi ]}$$. That is, in the countable case, we can add finitely many connectives to code the quantified variables in Theorem 1. Our last interpolation theorem addresses certain (natural) expansions of Fmr. We also give counterexamples showing that the scope of our results is the best possible. Our treatmet is algebraic via reducts of polyadic equality algebras. (2000 Mathematics Subject Classification. Primary 03G15. Secondary 03C05, 03C40.)