Cylindric algebras with terms

Abstract

In this paper we discuss cylindric algebras with terms. The setting is two—sorted algebras—one sort for terms and one for Boolean elements. As with cylindric algebras, a cylindric algebra with terms has its roots in first order predicate logic [HMT1].Let Σ be a set of sentences in a first order language with terms, equality and variables u0,u1,u2, …, Define a relation ≡Σ on Fm, the set of formulas, by φ ≡Σθ if and only if Σ ⊢ φ ↔ θ, and on Tm, the set of terms, by τ ≡Σσ if and only if Σ ⊢ τ ≈ σ. The operations +, ·, cκ, 0, 1 are defined as usual on equivalence classes. Define , where is σ with τ substituted for all occurrences of uκ. That the operation *κ, for κ < α, is well defined follows from the first order axioms of equality. Let vκ = [uκ]. To establish the link between terms and Booleans, define operations oκ as follows: , where φ' is a variant of φ such that uκ is free for τ in φ′ and is φ′ with τ substituted for all free occurrences of uκ in φ′. From the first order axioms it follows that oκ, for κ < α, is well defined. Finally, instead of diagonal elements, we define a Boolean-valued operation on terms as follows: [τ] e [σ] = [τ ≈ σ].