A Join Theorem for the Computably Enumerable Degrees
نویسندگان
چکیده
It is shown that for any computably enumerable (c.e.) degree w, if w 6= 0, then there is a c.e. degree a such that (a ∨w)′ = a′′ = 0′′ (so a is low2 and a ∨w is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low2 c.e. degrees are not elementarily equivalent as partial orderings.
منابع مشابه
Extensions of Embeddings below Computably Enumerable Degrees
Toward establishing the decidability of the two quantifier theory of the ∆ 2 Turing degrees with join, we study extensions of embeddings of upper-semi-lattices into the initial segments of Turing degrees determined by computably enumerable sets, in particular the degree of the halting set 0. We obtain a good deal of sufficient and necessary conditions.
متن کاملThe Non-isolating Degrees Are Upwards Dense in the Computably Enumerable Degrees
The existence of isolated degrees was proved by Cooper and Yi in 1995 in [6], where a d.c.e. degree d is isolated by a c.e. degree a if a < d is the greatest c.e. degree below d. A computably enumerable degree c is non-isolating if no d.c.e. degree above c is isolated by c. Obviously, 0 is a non-isolating degree. Cooper and Yi asked in [6] whether there is a nonzero non-isolating degree. Arslan...
متن کاملUniversal computably enumerable sets and initial segment prefix-free complexity
We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set A with non-trivial prefixfree initial segment complexity, there exists a Turing complete computably enumerable set B with complexity strictly less than the complexity of A. On the other hand it is known that s...
متن کاملEmbedding Finite Lattices into the Computably Enumerable Degrees — a Status Survey
We survey the current status of an old open question in classical computability theory: Which finite lattices can be embedded into the degree structure of the computably enumerable degrees? Does the collection of embeddable finite lattices even form a computable set? Two recent papers by the second author show that for a large subclass of the finite lattices, the so-called join-semidistributive...
متن کاملComputably Enumerable Reals and Uniformly Presentable Ideals
We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefix-free set of strings such that α = ∑ σ∈A 2 −|σ|. Note that ∑ σ∈A 2 −|σ| is precisely the measure of the set of reals that have a string in A as an initial segment. So we will simply abbreviate ∑ σ∈A 2 −|σ| by μ(A). It is know...
متن کامل