The recursive equivalence type of a decomposition of an ω-group: The RET of a decomposition

Abstract

Let ε, Λ, Λ*, ΛR stand for the set of non-negative integers, isols, isolic integers and regressive isols respectively, and let P(τ) be the ω-group of Gödel numbers of permutations of the set τ ε ε which move only finitely many elements of τ. The concept of an ω-group was studied by Hassett [5]. He proved in P12 of [5] that for an isolated set τ, the decomposition of P(τ) into conjugacy sets is a gc-decomposition if and only if τ is regressive. For the finite symmetric group on n elements, Sn it is known that the order of the conjugacy class is p(n), where p(n) is the partition function. The author shows in this paper, using a result of Barback [1], that if pΛ(T) is Nerode's canonical extension of p(n) to Λ and Req (τ) = T, then pΛ(T) = Req Cτ, where Cτ is the decomposition of P(τ) into conjugacy sets. The reader is assumed to be familiar with the contents of [2], [4] and [5].