EFFECTIVE INSEPARABILITY, LATTICES, AND PREORDERING RELATIONS

Abstract

AbstractWe 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.