Algebra and Logic

, Volume 35, Issue 2, pp 80–85 | Cite as

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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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • A. N. Dyogtev

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