each weak recursive degree possess a weak recursive strong

minimal cover (s.m.c.). So any finite linear ordering can be

imbedded as initial segment into weak recursive degrees.

Besides, since the class of r.e. array nonrecursive (a.n.r.) degrees

defined by Downey, Jockusch and Stob [1990] is complementary to

the class of weak recursive degrees in the r.e. degrees {\bf R}

and no a.n.r.degree can possess a s.m.c. we obtain as a corollary

that a r.e. degree possess a s.m.c. iff it is weak recursive.

A degree {\bf m} is a strong minimal cover for a (Turing)

degree {\bf a}, if {\bf a$<$m} and for each degree

{\bf c,\, c$<$m$\rightarrow$ c$\le$a}.

Spector in [1956] raised the problem of a decription of

degrees possessing s.m.c. but there was a little progress

in this direction. On the other hand, more success was

achieved in the description of

degrees that cannot be s.m.c. Clearly, all r.e.degrees and all

degrees relatively enumerable in a less degree are splittable

so they cannot be s.m.c. For example, the class

of all degrees greater than or equal to {\bf 0}$'$ and the class of

1-generic degrees are such.

In [1990] and [1996] Downey, Jockusch and Stob defined a notion

of array nonrecursive degree as such a degree {\bf a} that for any

function $f\le_{tt}K$, where $K$ is a creative set, there is a function $g$

recursive in {\bf a} such that $g(n)\ge f(n)$ for infinitely many $n$.

They shown that any degree {\bf a} from $\overline{GL_2}$ (i.e.satisfying

{\bf a$''>($a$\cup$0}$')'$) is a.n.r. but there are also low and low$_2$

a.n.r. degrees.

The most important is that a.n.r.degrees are closed upwards, splittable and

cupped to all higher degrees (more generally, any recursively presented

lattice with distinct least and greatest elements can be embedded in the

lower cone {\bf D$(\le$a}) bounded by an a.n.r. degree {\bf a} preserving

least and greatest elements and lattice operations, Fejer [1989],

Downey, Jockusch and Stob [1996]).

So no a.n.r.degree {\bf a} can be a s.m.c. or possess a s.m.c.

Decreasing a gap between a.n.r.degrees and degrees possessing s.m.c

we introduce a notion of weak recursive degrees and show that

any weak recursive degree has a s.m.c.

\vspace{1mm}

{\bf Definition} A degree {\bf a} is weak recursive, if there is

a recursive function $p(x)$ such that for every function

$f$ recursive in {\bf a} there is a weak array $\{W_{h(n)}\}_{n=0}^{\infty}$

such that for every $n\in\omega$ :

1. $|W_{h(n)}|\le p(n)$,

2. $f(n)\in W_{h(n)}$,

3. $x\ne y\,\rightarrow W_{h(x)}\cap W_{h(y)}=\emptyset$.

It follows immediately from the definition that the class of weak

recursive degrees $WR$ is closed downwards, i.e. for any degrees

{\bf a,b}, if {\bf a$<$b} and {\bf b}$\in WR$, then {\bf a}$\in WR$.

If a weak recursive degree {\bf a} is located below {\bf 0}$'$,

then every function $f$ recursive in {\bf a} is $p$-r.e.

i.e. has a recursive

approximation $f(s,n)$ such that for every $n$

$|\{f(s,n):\,s\in\omega\}|\le p(n)$.

Since a r.e.degree {\bf a} is $p$-r.e. for some increasing recursive

function $p$ iff it is not array nonrecursive (Downey, Jockusch

and Stob [1990]), we obtain that the class of r.e. weak recursive degrees

is complementary to the class of array nonrecursive degrees

in {\bf R}. So the following proposition holds:

{\bf Proposition}. Let {\bf a} be a r.e.degree. Then the

following conditions are equivalent:

(1) {\bf a} is weak recursive,

(2) {\bf a} is not array nonrecursive,

(3) Any {\bf b,\,b$\le$ a} has a strong minimal cover,

(4) There is a degree {\bf b$\ge a$} which is not the sup of a minimal

pair (in all degrees) (Downey [1993],\,Theorem 1.1),

(5) There is a non-splittable degree {\bf b$\ge a$},

This gives a partial answer to the question of

Downey, Jockusch and Stob [1996] about definability of the class of

a.n.r.degrees by showing that the class of r.e. a.n.r.degrees is definable

(in all degrees) by any property listed in points (3)-(5) of the

Proposition using the fact that {\bf R} is definable

in all degrees (Cooper [ta]).

\hspace{3 cm} {\bf R e f e r e n c e s}

S.B. Cooper [1996] Strong minimal covers for recursively enumerable

degrees, Mathmatical Logic Quarterly, {\bf 42} (1996), pp.191-196

S.B. Cooper [ta] On a conjecture of Kleene and Post, to appear

R. Downey [1993] Array nonrecursive degrees and lattice embeddings of

the diamond, III. J. Math. {\bf 37} (1993), pp.349-374

R. Downey, C. Jockusch, and M. Stob [1990]. Array nonrecursive

sets and multiple permitting arguments, in Recursion Theory

Week, Proceedings, eds. K. Ambow-Spies, G.H. Muller, and

G. Sacks, Lect. Notes Math.,v. 1432, Springer-Verlag, Berlin, 1990,

pp.141-173

R. Downey, C. Jockusch, and M. Stob [1996]. Array nonrecursive sets and

genericity, in Computability, Enumerability, Unsolvability: Directions in

Recursion Theory, eds S.B. Cooper, T.A. Slaman, S.S. Wainer, London

Mathematical Society, Lect. Notes Swries N 224, Cambridge University Press,

pp.93-104

P. Fejer [1989] Embedding lattices with top preserved below non-$GL_2$

degrees, Zeitschr. f.math. Logik und Grundlagen d.Math. {\bf 35}

(1989), 527-539

Sh. Ishmukhametov [ta1] Minimal covers for recursively enumerable

degrees, J.Symb.Log. to appear

Sh.T. Ishmukhametov [ta2] On a new method for constructing minimal

Turing degrees, Izv. Vyssh. Ucheb. Zaved. (Russian), to appear

P. Odifreddi [1989] Classical Recursion Theory, Studies in Logic and

the Foundations of Mathematics, Vol. 125, North-Holland, Amsterdam.

C.Spector [1956] On degrees of recursive unsolvability, Ann. of Math.

(2) {\bf 64}, pp.581-592

C.E.M. Yates [1970] Initial segments of the degrees of unsolvability,

Part II: Minimal degrees, J.Symb.Log. 35 (1970) pp.243-266