Strongly η-representable degrees and limitwise monotonic functions

Abstract

It is proved that each strongly η-representable degree contains a set that is a range of values for some 0 -limitwise monotonic function pseudoincreasing on Q. Thus we obtain a description of strongly η-representable degrees in terms of 0 -limitwise monotonic functions. PRELIMINARIESWe deal with problems arising at the junction of computability theory and the theory of linear orders. The notation and definitions of computability theory are borrowed from [1]. The set of natural numbers {0, 1, 2, . . .} is denoted by ω. We write X − Y for the set-theoretic difference between sets X ⊆ ω and Y ⊆ ω and write X for the complement ω − X of a set X ⊆ ω. If f is some function, then rang(f ) = {y | (∃x)[f (x) = y]} is its range. Let f : A × ω −→ ω; then lim s→∞ f (x, s) is defined and is finite if there exists a number a x such that (∃s 0 )(∀s > s 0 )[f (x, s) = a x ].By writing x . y we mean a bounded difference; namely, x . y = x − y if x y and x . y = 0 otherwise. We fix an arbitrary computable bijective function N : ω −→ Q and define q i = N (i) for any i ∈ ω.A linear ordering L = L, < L is said to be computable if the universe L and the order relation < L are computable. A standard order relation on ω is denoted by <. For an order type of a dense linear ordering with no greatest and least elements, we write η. The set of rational numbers is denoted by Q.

The Simplest Low Linear Order With No Computable Copies

“…Frolov and Zubkov [24] and Turetsky and Kach [20], independently, proved that a degree is increasing -representable if and only if it is -computable. We will note here only one more result obtained by Zubkov [22]. A degree is strongly -representable if and only if it has the range of a -limitwise monotonic function which satisfies the following conditions: for all x , the set of x such that is ordered as , and if and then .…”