# Closures

• Lane A. Hemaspaandra
• Leen Torenvliet
Part of the Monographs in Theoretical Computer Science An EATCS Series book series (EATCS)

## Abstract

In this chapter we will study several closure properties of the P-selective sets. That is, for various functions h we will ask whether h(A 1,..., A k ) is P-selective whenever all the A i ’s are P-selective. Theorem 1.8 states that for complementation (that is, k = 1 and h(A 1) = Ā1) the answer is yes; the complement of a P-selective set is always P-selective. In this chapter we will be in part concerned with closure under boolean operations, i.e., with functions of the form χh(A 1..., A k ) = fA 1, ... ,χA k ) for some boolean function ƒ. In particular, we will study so-called boolean connectives. A boolean connective I(ƒ) is defined using a boolean operator ƒ; I(ƒ) works on a collection of sets and returns a single set using, in the fashion just described, the boolean operator ƒ with which it is defined. For instance, let ƒ or denote the boolean function such that ƒ or (a, b) = ab. The connective Ior) works on two sets A and B and, since it applies an element-by-element “or,” returns AB (i.e., I(f or) (A, B) = AB). Indeed, a boolean connective I(ƒ) can be defined for any boolean function ƒ. Since there are $${2^{{2^k}}}$$ boolean functions on k variables, it follows that there are $${2^{{2^k}}}$$ potential boolean closures to be investigated for the P-selective sets. Section 5.2 will give a complete account of this investigation. In particular, for each k we will determine under exactly how many—and which—k-ary boolean connectives the P-selective sets are closed.

### Keywords

Sorting Prefix Hema TanOl