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.