Algebra and Logic, Vol. 40, No. 5, 2001 ROGERS SEMILATTICES OF FAMILIES OF ARITHMETIC SETS S. A. Badaev and S. S. Goncharov * UDC 510.5 We look into algebraic properties of Rogers semilattices of arithmetic sets, such as the existence of minimal elements, minimal covers, and ideals without minimal elements. The article deals with algebraic properties of Rogers semilattices of families of arithmetic sets, such as the existence of minimal elements, minimal covers, and ideals without minimal elements. The notion of a computable numbering of a family of arithmetic sets was introduced by Goncharov and Sorbi in the framework of their general approach to defining the notion of computability for families of constructive objects admitting a formal description in some language endowed with a given G¨odel numbering of formulas (cf. [1]). The derived notion of a Rogers semilattice of computable numberings of a family allows us to establish algebraic interconnections among different ways of computing the family. 1. BASIC NOTIONS AND THE NOTATION We refer the reader to [2-4] for the relevant notions and the standard notation in algorithm theory and in the theory of numberings. Let A be a family of sets lying in some class Σ 0 n , n  1, of the arithmetic hierarchy. A numbering α : ω → A is said to be Σ 0 n -computable if {〈x, m〉| x ∈ α(m)}∈ Σ 0 n . Clearly, the notion of a Σ 0 1 -computable numbering coincides with the classical notion of a computable numbering of a family of computably enumerable (c.e.) sets in [4]. Using the Kleene–Post strong hierarchy theorem (cf. [3]), it was shown in [1] that the Σ 0 n+1 -computability of a numbering α means that the set {〈x, m〉| x ∈ α(m)} of pairs of α can be enumerated by a 0 (n) -computable function. Let Com 0 n (A) be the set of all Σ 0 n -computable numberings of a family A. A numbering α of A is reducible to a numbering β of that family (written α  β) if there exists a computable function f such that α(x)= β(f (x)) for all x ∈ ω. If α  β and β  α then the numberings α and β are said to be equivalent (written α ≡ β). Denote by deg(α) the degree of α, that is, the set {β | β ≡ α} of numberings. The reducibility of numberings is a pre-order relation on Com 0 n (A), and it induces a partial-order relation on a set of degrees of the numberings in Com 0 n (A), which we also denote by . The partially ordered set R 0 n (A) ⇋ 〈{deg(α) | α ∈ Com 0 n (A)}, 〉 is an upper semilattice, which we call the Rogers semilattice of the family A. * Supported by INTAS–RFFR grant No. 97-139. Translated from Algebra i Logika, Vol. 40, No. 5, pp. 507-522, September-October, 2001. Original article submitted October 11, 2000. 0002-5232/01/4005-0283 $25.00 c  2001 Plenum Publishing Corporation 283