Recognizable algebras of formulas.

Abstract

In this paper we consider various algebras formed out of the formulas of first-order languages. We rely mostly on [l] for our notations and terminology. We deal with structures, 31 = (A, Rθ)θ ψ} usually we just write \φ\. In general we present our results for a collection of languages at a time; in particular, _yV = {-C } and Ji = {-Caa}. Note that JiQjsl and £ωω is the usual finitary first-order language with equality. Unless otherwise specified we assume that <£e Js/. The notions of elementary equivalence, elementary extension, and elementary embedding can be extended to the infinitary languages and we write -C Ξ , «C -3, and ^-embedding respectively. We also deal with second-order languages £; j£ contains all the symbols of <£ and variables of every degree y, 0 < γ < α, which we write as vj. In the model under consideration the vj are interpreted as variable y-ary relations. In £ we may quantify over such relations. Individual variables have degree 0 and are denoted by x, y, and z.