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.

中文翻译：

---

有效不可分性、格和预定关系

我们有效地研究不可分（缩写为 ei）的前格（即，形式的结构$L = \langle \omega , \wedge , \vee ,0,1,{ \le _L}\rangle$在哪里ω表示自然数集，以下四个条件成立： (1)$\楔形, \vee$是二进制可计算操作；(2)${ \le _L}$是一个可计算的可枚举预排序关系，其中$0{ \le _L}x{ \le _L}1$对于每个X; (3)等价关系${ \equiv _L}$起源于${ \le _L}$是一致的大号使得相应的商结构是一个非平凡的有界格；(4)${ \equiv _L}$- 0 和 1 的等价类形成一对有效不可分的集合）。解决（Montagna & Sorbi, 1985）中的一个问题，我们证明（定理 4.2），如果大号那么是一个ei prelattice${ \le _L}$对于所有 ce 前序关系是普遍的，即对于每个 ce 前序关系R存在一个可计算的函数F减少R到${ \le _L}$， IE，$xRy$当且仅当$f\left( x \right){ \le _L}f\left( y \right)$， 对所有人$x,y$. 事实上（推论 5.3）${ \le _L}$是局部普遍的，即，对于每一对$a{ < _L}b$和每一个 ce 前序关系R可以找到一个归约函数F从R到${ \le _L}$这样的范围F包含在区间内$\left\{ {x:a{ \le _L}x{ \le _L}b} \right\}$. 也（定理 5.7）${ \le _L}$是均匀稠密的，即存在一个可计算的函数F这样对于每个$a,b$如果$a{ < _L}b$然后$a{ < _L}f\left( {a,b} \right){ < _L}b$， 而如果$a{ \equiv _L}a\prime$和$b{ \equiv _L}b\prime$然后$f\left( {a,b} \right){ \equiv _L}f\left( {a\prime ,b\prime } \right)$. 讨论了这些结果的一些后果和应用：特别是（推论 7.2）$n \ge 1$ce 的前序关系${{\rm{\Sigma }}_n}$由鲁滨逊系统的任何一致性扩展的可证明蕴涵关系产生的句子R要么问是局部普遍且均匀稠密的；和（推论 7.3）由 Heyting 算术的任何 ce 一致扩展的可证明暗示所产生的 ce 预排序关系是局部普遍且均匀密集的。