Various notions of represetability for cylindric and polyadic algebras

For β an ordinal, let PEAβ (SetPEAβ) denote the class of polyadic equality (set) algebras of dimension β. We show that for any infinite ordinal α, if is atomic, then for any n < ω, the n-neat reduct of , in symbols , is a completely representable PEAn (regardless of the representability of ). That is to say, for all non-zero , there is a and a homomorphism such that fa(a) ≠ 0 and for any for which exists. We give new proofs that various classes consisting solely of completely representable algebras of relations are not elementary; we further show that the class of completely representable relation algebras is not closed under ≡∞,ω. Various notions of representability (such as ‘satisfying the Lyndon conditions’, weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CAns satisfying the Lyndon conditions coincides with the class of atomic algebras in ElScNrnCAω, where El denotes ‘elementary closure’ and Sc is the operation of forming complete subalgebras.

Constant Flux States in Anisotropic Turbulence

An elementary closure theory is used to compute the scaling of anisotropic contributions to the correlation function in homogeneous turbulence. These contributions prove to decay with wavenumber more rapidly than the energy spectrum; this property is sometimes called the “recovery of isotropy” at small scales and is a key hypothesis of the Kolmogorov theory. Although comparisons with a more comprehensive theory suggest that the present theory is too crude, its elementary character makes the scaling analysis straightforward. The analysis reveals some characteristic features of anisotropic turbulence, including “angular” energy transfer in wavevector space.

Functionals defined by transfinite recursion

This paper deals mainly with quantifier-free second order systems (i.e., with free variables for numbers and functions, and constants for numbers, functions, and functionals) whose basic rules are those of primitive recursive arithmetic together with definition of functionals by primitive recursion and explicit definition. Precise descriptions are given in §2. The additional rules have the form of definition by transfinite recursion up to some ordinal ξ (where ξ is represented by a primitive recursive (p.r.) ordering). In §3 we discuss some elementary closure properties (under rules of inference and definition) of systems with recursion up to ξ. Let Rξ denote (temporarily) the system with recursion up to ξ. The main results of this paper are of two sorts:Sections 5–7 are concerned with less elementary closure properties of the systems Rξ. Namely, we show that certain classes of functional equations in Rη can be solved in Rη for some explicitly determined η < ε(η) (the least ε-number > ξ). The classes of functional equations considered all have roughly the form of definition by recursion on the partial ordering of unsecured sequences of a given functional F, or on some ordering which is obtained from this by simple ordinal operations. The key lemma (Theorem 1) needed for the reduction of these equations to transfinite recursion is simply a sharpening of the Brouwer-Kleene idea.