Effective Inseparability, Lattices, and Preordering Relations

Abstract

We study effectively inseparable (e.i.) pre-lattices (i.e. structures of the form L " xω,^, _, 0, 1, ďLy where ω denotes the set of natural numbers and the following hold:^, _ are binary computable operations; ďL is a c.e. pre-ordering relation, with 0 ďL x ďL 1 for every x; the equivalence relation "L originated by ďL is a congruence on L such that the corresponding quotient structure is a non-trivial bounded lattice; the "L-equivalence classes of 0 and 1 form an effectively inseparable pair), and show (Theorem 30, solving a problem in [17]), that if L is an e.i. pre-lattice then ďL is universal with respect to all c.e. pre-ordering relations, i.e. for every c.e. pre-ordering relation R there exists a computable function f such that, for all x, y, x R y if and only if f pxq ďL f pyq; in fact (Corollary 43) ďL is locally universal, i.e. for every pair a ăL b and every c.e. pre-ordering relation R one can find a reducing function f from R to ďL such that the range of f is contained in the interval tx : a ďL x ďL bu. Also (Theorem 47) ďL is uniformly dense, i.e. there exists a computable function f such that for every a, b if a ăL b then a ăL f pa, bq ăL b, and if a "L a 1 and b "L b 1 then f pa, bq "L f pa 1 , b 1 q. Some consequences and applications of these results are discussed: in particular (Corollary 55) for n ě 1 the c.e. pre-ordering relation on Σn sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson's Q or R is locally universal and uniformly dense; and (Corollary 56) the c.e. pre-ordering relation of provable implication of Heyting Arithmetic is locally universal and uniformly dense.2010 Mathematics Subject Classification. 03D25.

Word problems and ceers

“…ring whose word problem is strongly isomorphic to . The following result is essentially a rephrasing of Theorem 4.1 of [5]. Lemma Let A be a c.e.…”

“…Proof For the convenience of the reader, we recall the argument in [5], adapting it to our context and notations. We look for a computable function such that if , and for some then .…”

“…P r o o f . For the convenience of the reader, we recall the argument in[5], adapting it to our context and notations. We look for a computable function f (D, e, x) such that if ϕ e (x) ↓, and ϕ e (x) = A d for some d ∈ D then f (D, e, x) = A ϕ e (x).Let p be a productive function for the pair (U 0 , U 1 ): it is well known that we may assume that p is total.Let {u d,D,e,x , vd,D,e,x : D is a finite subset of ω, d ∈ D, e, x ∈ ω} be a computable set of indices we control by the Recursion Theorem.…”