Algebra and Logic

, Volume 35, Issue 2, pp 80–85

# Implicatively selector sets

• A. N. Dyogtev
Article

## Abstract

Let A⊆N={0,1,2,...} and β be an n-ary Boolean function. We call A a β-implicatively selector (β-IS) set if there exists an n-ary selector general recursive function f such that (∀x1,...,xn)(β(χ(x1),...,χ(xn))=1⟹f(x1,...,xn)∈A), where χ is the characteristic function of A. Let F(m), m≥1, be the family of all d m+1 * -IS sets, where$$d_{m + 1}^* = \mathop \& \limits_{1 \leqslant i< j \leqslant m + 1} (x_i \vee x_j )$$, F(0)=N, and F(∞) is the class of all subsets in N. The basic result of the article says that the family of all β-IS sets coincides with one of F(m), m≥0, or F(∞), and, moreover, the inclusions F(0)⊂F(1)⊂...⊂F(∞) hold.

### Keywords

Characteristic Function Mathematical Logic Boolean Function Basic Result Recursive Function

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### References

1. 1.
A. N. Dyogtev, “Recursively-combinatorial properties of subsets of natural numbers,”Algebra Logika,29, No. 3, 303–314 (1990).Google Scholar
2. 2.
S. V. Yablonskii, G. P. Gavrilov, and V. G. Kudryavtsev,Functions of the Algebra of Logic and Post Classes [in Russian], Nauka, Moscow (1966).Google Scholar
3. 3.
C. E. M. Yates, “Recursively enumerable sets and retracing functions,”Z. Math. Log. Grund. Math.,8, No. 4, 331–345 (1962).
4. 4.
C. G. Jockusch, “Semirecursive sets and positive reducibility,”Trans. Am. Math. Soc.,131, No. 2, 420–436 (1968).