Partially ordered sets of 1-degrees, contained in recursively enumerable m-degrees

Abstract

If A is a recursively enumerable (r.e.) nonrecursive set, then the family of all 1-degrees, contained in the m-degree of the set A, along with the natural relation ~ for the ldegrees, forms a partially ordered set L(A). In the present article, we investigate the structure of L(A) in terms of the properties of the set A. Let N={O,4... I" If A ~N, then f-N\A, and JA[ is the power of the set A. We denote by pf and ~f the domain of values and domain of definition of the one-place function f. We recall that, given any A, L(AI contains a maximal element. We will call aeL(A) a minimal element if it is not the least element in L(A), and, given any ~eL(A), ~ implies ~=a or that b is the least element of L(A). The concept of maximal element is defined in a similar way. We shall say that L(~) is dense if (~a)(~)(O,~EL(~)~a~==,~(3C)(a~c~). If L(A)con~ slats of a single element, then the m-degree of the set A is called irreducible. In this case, the m-degree of A contains only cylinders. It may be mentioned in this connection that simple and pseudosimple sets are not cylinders [i]. Some facts that are easily extracted from familiar results will be stated as propositions. Proposition I. If the m-degree of an r.e. set is not irreducible, then L(A) has no maximal elements and contains both an infinite chain and an infinite number of pairwise noncomparable elements. This follows from the results of [2, 3]. For, it was shown in [2] that, if A is not a cylinder, then A~A is likewise not a cylinder, and ~t~@A 9 When constructing the antichain, essential use was made of the recursive enumerability of the set A [3]. Proposition 2. If L(AJ contains minimal elements, then L(A) has a least element. This is obvious in the case when L(A} contains only one minimal element. If A ~ ~, ~m ~z, the 1-degrees of the sets Bz and B2 are minimal and noncomparable, and ~mBz by means of the general recursive function (g.r.f.) ~ , then let