Embedding Lattices with Top Preserved Below Non-GL2 Degrees

A minimal degree not realizing least possible jump. A recursively enumerable degree that is not the top of a diamond in the Turing degrees. 18 6] Peter A. Fejer and Richard A. Shore. Embeddings and extensions of embeddings in the r.e. tt-and wtt-degrees. 17 Corollary 8 If a is not in GL 2 , then any nontrivial recursively presented lattice can be embedded (as a lattice) into D (a) preserving least and greatest element, if they exist. Corollary 9 If a is not in GL 2 , then any nite nontrivial partial order can be embedded into D (a), preserving least and greatest elements, if they exist, and all joins and infs that exist in the partial order. Corollary 10 If a is not in GL 2 , then the 9 theory of the structure D (a) in the language L = (; _; ^; 0; 1) is decidable. Proof: This proof is essentially as in Fejer-Shore 6], Corollary to Theorem 3. That is, given an 9 sentence = 9x 1 : : : 9x n in L with quantiier free (where _ is taken to be a three place relation symbol), we claim that is valid in D (a) if and only if is valid in some nontrivial bounded partial order with at most 2(n + 2) 3 + (n + 2) elements. We refer the reader to 6] for details. must be some t t 0 such that g t has at least n zeros, and hence g has at least n zeros. Now take s 1 s 0 such that g s 1 has at least n zeros and let s s 1 be such that h(s) F(s). (This is possible since F does not dominate h.) Then by the construction, either there is an attack made on R n before stage s + 1 which is not canceled by the end of stage s and is not discredited at stage s + 1 or else an attack is made on R n at stage s + 1. Let t + 1 s + 1 be the stage at which the attack on R n which exists at the end of stage s + 1 was made. If, during the attack at stage t + 1 on R n , a search was made for a string or strings of length h(t) and no such string(s) were …