The class of polyadic algebras has the super amalgamation property

Abstract

We show that for infinite ordinals α the class of polyadic algebras of dimension α has the super amalgamation property.The most well known generic examples of algebraisations of first order logic are Tarski's cylindric algebras and Halmos' polyadic algebras. Both examples are well known and are widely used. In both cases, it turns out that the locally finite algebras are representable, and this is equivalent to the completeness of first order logic. While there are cylindric algebras of infinite dimension that are not representable, Daigneault and Monk prove a strong extension of the representation theorem for locally finite polyadic algebras (due to Halmos) namely, every polyadic algebra of infinite dimension (without equality) is representable [4]. This is a point where the two theories deviate. The latter, a typical Stone-like representation theorem, reflects the fact that the notion of polyadic algebra is indeed an adequate reflection of Keisler's predicate logic (KL). KL is a proper extension of first order logic without equality, obtained when the bound on the number of variables in formulas is relaxed; and accordingly allowing the following as extra operations on formulas: Quantification on infinitely many variables and simultaneous substitution of (infinitely many) variables for variables. By the representation theorem in [4], KL is complete. Keisler gave an independent (meta-mathematical proof) of this fact.It is quite natural to ask for algebraic versions of model theoretic results, other than completeness, reflected algebraically by Stone-like representability results. Such results could concern for example interpolation theorems, or omitting types theorems Omitting types theorems for KL proves problematic, since KL is necessarily "uncountable", while omitting types theorems are very much tied to countability. However, Daigneault [3] succeeded in stating and proving versions of Beth's and Craig's theorems by proving that locally finite polyadic algebras (with and without equality) have the amalgamation property. Later Johnson removed the condition of local finiteness, proving that polyadic algebras of infinite dimension without equality have the strong amalgamation property [5]. Here we strengthen Johnson's result proving that the class of polyadic algebras has the super amalgamation property. The super amalgamation property (which follows) was introduced by Maksimova [14].Definition 1 Let W be a class of algebras each having a Boolean reduct. W has the super amalgamation property, or SUPAP for short, if for all A 0 , A 1 , A 2 ∈ W and all monomorphisms i 1 and i 2 of A 0 into A 1 , A 2 , respectively, there exists A ∈ W , a monomorphism m 1 from A 1 into A and a monomorphism m 2 from A 2 into A such that m 1 • i 1 = m 2 • i 2 , and for all x ∈ A j and for all y ∈ A k , if m j (x) ≤ m k (y), then there exists z ∈ A 0 such that x ≤ i j (z) and i k (z) ≤ y where {j, k} = {1, 2}.For an algebra A and X ⊆ A, Sg A X or simply Sg X denotes the subalgebra of A generated by X. The next definition is an algebrai...