Σ-Bounded algebraic systems and universal functions. I

Abstract

We introduce the concept of a Σ-bounded algebraic system and prove that if a system is Σ-bounded with respect to a subset A then in a hereditarily finite admissible set over this system there exists a universal Σ-function for the family of functions definable by Σ-formulas with parameters in A. We obtain a necessary and sufficient condition for the existence of a universal Σ-function in a hereditarily finite admissible set over a Σ-bounded algebraic system. We prove that every linear order is a Σ-bounded system and in a hereditarily finite admissible set over it there exists a universal Σ-function. Now it is generally accepted that one of the important generalizations of the concept of computability is Σ-definability (generalized computability) in admissible sets. This generalization has made it possible to study computability problems over arbitrary algebraic systems, for instance, over the field of real numbers. The most important results of computability theory in admissible sets and their applications to theoretical computer science (semantic programming, dynamic logic, the theory of effective f -spaces, and so on) are collected in the monograph of Ershov [1].One of the principal results of absolute computability theory is the existence of a universal partially computable function. It is known (see [1]) that in every admissible set there exists a universal Σ-predicate; however, this is false for Σ-functions. An algebraic system M is constructed in [2] such that in the hereditarily finite admissible set HF(M) there is no universal Σ-function. Therefore, it is interesting to know which conditions on M guarantee the existence of a universal Σ-function in the hereditarily finite admissible set HF(M) over M. It is proved in [1] that if M is an algebraic system of a decidable and model complete theory then in HF(M) there exists a universal Σ-function. For a certain class K of algebraic systems, necessary and sufficient conditions for the existence of a universal Σ-function in HF(M), where M ∈ K, were found in [3][4][5]. A torsion-free abelian group A was constructed in [6] such that in HF(A) there is no universal Σ-function.In this article we introduce the concept of a Σ-bounded (with respect to a finite subset) algebraic system. We prove that if M is a Σ-bounded system with respect to a finite subset M 0 then in HF(M) there exists a universal Σ-function for the family of all Σ-functions determined by Σ-formulas with parameter M 0 . We obtain a necessary and sufficient condition for the existence of a universal Σ-function in the hereditarily finite admissible set over a Σ-bounded algebraic system. We prove that every linear order is a Σ-bounded system and in the hereditarily finite admissible sets over it there exist universal Σ-functions. As regards terminology and notation for admissible sets, we follow [1]. Also, we use the following notation. Take an algebraic system M of a finite signature σ 0 , whose underlying set is denoted by M , and some subset M 0 ⊆ M . Put σ 1 = {σ 0 , ∈, U 1 , ∅} and σ i (M 0 ) = σ i ∪...

Universal Functions Over Trees

“…In [4], a torsion-free Abelian group A was constructed for which HF(A) has no universal Σ-function. In [5,6], it was proved that a universal Σ-function exists in hereditarily finite superstructures over an Abelian p-group, a linear order, and an Ershov algebra. In [7,8], the concept of a Σ-uniform structure was introduced and a condition was specified that is necessary and sufficient for a universal Σ-function to exist *…”

“…Definition 2 [5]. Suppose that a locally constructivizable structure M and its finite subset M 0 satisfy the following conditions:…”

Σ-bounded algebraic systems and universal functions. II

“…This article continues [1] where we had introduced the concept of a Σ-bounded algebraic system and obtained a necessary and sufficient condition for the existence of universal Σ-functions in a hereditarily finite admissible set over a Σ-bounded system. In this article we prove that Ershov algebras, Boolean algebras, and abelian p-groups are Σ-bounded systems, and universal Σ-functions exist over them.…”

“…In particular, every two bases of the same characteristic are of the same length. Theorem A [1,Corollary 4]. If an algebraic system M is Σ-bounded with respect to a finite subset M 0 of M then there exists a universal Σ-function U M 0 (x, y) ∈ F Σ(HF(M), M 0 ) for the family F M 0 such that every f ∈ F M 0 satisfies λyU M 0 (n, y) = f (y) for some n.…”

“…Theorem B [1,Theorem 2]. Given a Σ-bounded algebraic system M, in HF(M) there exists a universal Σ-function with parameter A if and only if for every finite subset C with respect to which M is Σ-bounded there is a finite subset C 1 such that for every finite subset X and every base…”