18. A. I. Zimin, "Semigroup varieties with finiteness conditions for finitely generated semi- groups," Dep. at VINITI, 27.10.1980, No. 4519-80. 19. S. I. Kublanovskii, "Locally residually finite and locally representable varieties of associative rings and algebras," Dep. at VINITI, 14.12.1982, No. 6143-82. ERSHOV HIERARCHY AND THE T-JUMP V. L. Selivanov UDC 510.5 %=, -t -I Let IO,~<O) be a Kleene system of ordinal notation [i], ~.J= , ~= classes of the Ershov hierarchy [2, 3], and ' the operation of the T-jump. The basic purpose of this paper is the proof of the following theorem. THEOREM i. i) For any recursively enumerable (r.e.) set A and any nonleast a ~ 0 there --~! , ~' ~/. 2) For any exists an ,~ £~__ not T-equivalent to any set from such that ~_/ r.e. set A and any limit a e 0 there exists an ~ , not T-equivalent to any set from -I I l ~<~a~{ , such that 2 ~TA, This theorem completely describes jumps of sets from the Ershov hierarchy, since we noted -/ [4] that if a • 0 is a successor for b • 0, then any A~1-set is T-equivalent to some Z/ -set. For the case of differences of r.e. sets forming the second level of Ershov hierarchy, the theorem was proved by S. T. Ishmukhametov [5]. In his proof he made much use of the specifics of differences, which is not suitable, in our view, for other levels. Our proof is based on completely different ideas and is about three times shorter. Theorem 1 implies the following assertion, which informally denotes the "independence" of the hierarchy of higher and lower degrees and the hierarchy of degrees induced by the Ershov hierarchy. Let Ln be the set of all T-degrees d~o ! such that ~ =6/ )//=L~-~O/I~ = O~"+"j,S=Ld-<O'I ¥~(d¢iA$¢i~] Let ~ and ~ be aggregates of T-degrees containing sets from ~/ and zj~ / respectively, % 4 sf 4 :: COROLLARY. i) For any n • N and nonleast a e 0 the classes ~ +4' ~ ~ and ~/~ are nonempty. 2) For any n • N and limit a ~ 0 the classes ~ nL~+,,_= ~, and ~ n ~/ are nonempty. Let us pass to the proof of the theorem. We will introduce some notation. We identify sets with characteristics functions, i.e., n • X and X(n) = i, as well as n ~ N and X(n) = 0, are equivalent. Let X~t~>l <~x>eX~, X~{<8,~6X ~<~j. ~(~)~(~(~)~.) denotes the partial function ~ is determinate (indeterminate) at the point n, and @[n] denotes the re- strictions of ~ to the set {xlx < n}. If X is a r.e. set ~ $ ) , then X s is a finite set calculated in s steps in some effective enumeration of X (or in some limit calculation for Translated from Algebra i Logika, Vol. 27, No. 4, pp. 464-478, July-August, 1988. Orig- inal article submitted December 24, 1986. 292 0002-5232/88/2704-0292512.50 © 1989 Plenum Publishing Corporation