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EFFECTIVE INSEPARABILITY, LATTICES, AND PREORDERING RELATIONS

Part of: Computability and recursion theory

Published online by Cambridge University Press:  12 July 2019

URI ANDREWS  and
ANDREA SORBI
URI ANDREWS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSINMADISON, WI53706-1388, USAE-mail: andrews@math.wisc.eduURL: http://www.math.wisc.edu/andrews/
ANDREA SORBI
Affiliation:
DIPARTIMENTO DI INGEGNERIA DELL’INFORMAZIONE E SCIENZE MATEMATICHE UNIVERSITÀ DI SIENASIENA, I-53100, ITALYE-mail: andrea.sorbi@unisi.itURL: http://www3.diism.unisi.it/sorbi/
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Abstract

We study effectively inseparable (abbreviated as e.i.) prelattices (i.e., structures of the form $L = \langle \omega , \wedge , \vee ,0,1,{ \le _L}\rangle$ where ω denotes the set of natural numbers and the following four conditions hold: (1) $\wedge , \vee$ are binary computable operations; (2) ${ \le _L}$ is a computably enumerable preordering relation, with $0{ \le _L}x{ \le _L}1$ for every x; (3) the equivalence relation ${ \equiv _L}$ originated by ${ \le _L}$ is a congruence on L such that the corresponding quotient structure is a nontrivial bounded lattice; (4) the ${ \equiv _L}$ -equivalence classes of 0 and 1 form an effectively inseparable pair of sets). Solving a problem in (Montagna & Sorbi, 1985) we show (Theorem 4.2), that if L is an e.i. prelattice then ${ \le _L}$ is universal with respect to all c.e. preordering relations, i.e., for every c.e. preordering relation R there exists a computable function f reducing R to ${ \le _L}$ , i.e., $xRy$ if and only if $f\left( x \right){ \le _L}f\left( y \right)$ , for all $x,y$ . In fact (Corollary 5.3) ${ \le _L}$ is locally universal, i.e., for every pair $a{ < _L}b$ and every c.e. preordering relation R one can find a reducing function f from R to ${ \le _L}$ such that the range of f is contained in the interval $\left\{ {x:a{ \le _L}x{ \le _L}b} \right\}$ . Also (Theorem 5.7) ${ \le _L}$ is uniformly dense, i.e., there exists a computable function f such that for every $a,b$ if $a{ < _L}b$ then $a{ < _L}f\left( {a,b} \right){ < _L}b$ , and if $a{ \equiv _L}a\prime$ and $b{ \equiv _L}b\prime$ then $f\left( {a,b} \right){ \equiv _L}f\left( {a\prime ,b\prime } \right)$ . Some consequences and applications of these results are discussed: in particular (Corollary 7.2) for $n \ge 1$ the c.e. preordering relation on ${{\rm{\Sigma }}_n}$ sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson’s system R or Q is locally universal and uniformly dense; and (Corollary 7.3) the c.e. preordering relation yielded by provable implication of any c.e. consistent extension of Heyting Arithmetic is locally universal and uniformly dense.

Keywords

computable reducibility on preorderseffective inseparabilityuniform finite precompletenessuniform density

MSC classification

Primary: 03D30: Other degrees and reducibilities
Secondary: 03D45: Theory of numerations, effectively presented structures
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 4 , December 2021 , pp. 838 - 865
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Copyright
© Association for Symbolic Logic 2019

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