Embedding lattices in the Kleene degrees

Under ZFC+CH, we prove that some lattices whose cardinalities do not exceed א1 can be embedded in some local structures of Kleene degrees. 0. We denote by 2E the existential integer quantifier and by χA the characteristic function of A, i.e. x ∈ A⇔ χA(x) = 1, and x 6∈ A⇔ χA(x) = 0. Kleene reducibility is defined as follows: for A,B ⊆ ω, A ≤K B iff there is a ∈ ω such that χA is recursive in a, χB , and 2E. We introduce the following notations. K denotes the upper semilattice of all Kleene degrees with the order induced by ≤K. For X,Y ⊆ ω, we set X ⊕ Y = {〈0〉 ∗ x | x ∈ X} ∪ {〈1〉 ∗ x | x ∈ Y }. Then deg(X ⊕ Y ) is the supremum of deg(X) and deg(Y ). The superjump of X is the set XSJ = {〈e〉 ∗x ∈ ω | {e}((x)0, (x)1, χX , 2E)↓}. Here, 〈e〉 ∗x is the real such that (〈e〉 ∗ x)(0) = e and (〈e〉 ∗ x)(n+ 1) = x(n) for n ∈ ω. More generally, for m ∈ ω, 〈e0, . . . , em〉 ∗ x is the real such that (〈e0, . . . , em〉 ∗ x)(n) = en for n ≤ m and (〈e0, . . . , em〉 ∗ x)(n + m + 1) = x(n) for n ∈ ω. Further, (x)0 = λn.x(2n) and (x)1 = λn.x(2n + 1). We identify 〈(x)0, (x)1〉 with x. An X-admissible set is closed under λx.ω 1 iff it is X SJ-admissible. The following conditions (1) and (2) are equivalent to A ≤K B ([8]). (1) There is y ∈ ω such that A is uniformly ∆1-definable over all (B; y)admissible sets; i.e. there are Σ1(Ḃ) formulas φ0 and φ1 such that for any (B; y)-admissible set M and for all x ∈ ω ∩M , x ∈ A⇔M |= φ0(x, y)⇔M |= ¬φ1(x, y). (2) There are y ∈ ω and Σ1(Ḃ) formulas φ0 and φ1 such that for all x ∈ ω, x ∈ A⇔ LωB;x,y 1 [B;x, y] |= φ0(x, y)⇔ LωB;x,y 1 [B;x, y] |= ¬φ1(x, y). 1991 Mathematics Subject Classification: 03D30, 03D65.