有效不可分性、格和预序关系
Effective inseparability, lattices, and pre-ordering relations
摘要：
我们有效地研究不可分割的（即）预格（即，形式为$L=\langle ω, \wedge, \lor, 0, 1, \leq_L\rangle$的结构，其中$ω$表示自然数的集合，并且以下成立：$\wedge, \lor$是二进制可计算运算; $\leq_L$是c.e.预序关系，其中$0 \leq_{L} x \leq_{L} 1$用于每个$x$ ;等价关系$\equiv_L$起源于$\leq_L$是一个同余$L$，使得相应的商结构是一个非平凡的有界格; $\equiv_L$ -等价类$0$和$1$形成一个有效的不可分对），并表明，如果$L$是一个e.i.预格，则$\le_{L}$对于所有c. e.预排序关系，即对于每个c.e.预排序关系$R$存在一个可计算函数$f$，使得对于所有$x,y$，$x \mathrel{R} y$当且仅当$f(x) \le_{L} f(y)$ ;事实上$\leq_L$是局部通用的，即对于每对$a<_{L} b$和每个c. e.预排序关系$R$可以找到从$R$到$\le_{L}$的归约函数$f$，使得$f$的范围包含在区间$\{x: a \leq_{L} x \leq_{L} b\}$中。$\leq_L$也是一致稠密的，即存在一个可计算函数$f$，使得对于每个$a,b$，如果$a<_{L} b$，则$a<_{L} f(a,b) <_{L} b$，如果$a\equiv_{L} a'$和$b \equiv_{L} b'$，则$f(a,b)\equiv_{L} f(a',b')$。这些结果的一些后果和应用进行了讨论：特别是$n \ge 1$的c. e。由任意c.e.的可证蕴涵关系所产生的$Σ_{n}$语句上的预序关系。Robinson的$Q$或$R$的相容扩张是局部泛的和一致稠密的; Heyting算术可证蕴涵的预序关系是局部泛的、一致稠密的。
英文摘要：
We study effectively inseparable (e.i.) pre-lattices (i.e. structures of the form $L=\langle ω, \wedge, \lor, 0, 1, \leq_L\rangle$ where $ω$ denotes the set of natural numbers and the following hold: $\wedge, \lor$ are binary computable operations; $\leq_L$ is a c.e. pre-ordering relation, with $0 \leq_{L} x \leq_{L} 1$ for every $x$; the equivalence relation $\equiv_L$ originated by $\leq_L$ is a congruence on $L$ such that the corresponding quotient structure is a non-trivial bounded lattice; the $\equiv_L$-equivalence classes of $0$ and $1$ form an effectively inseparable pair), and show that if $L$ is an e.i. pre-lattice then $\le_{L}$ 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 $x,y$, $x \mathrel{R} y$ if and only if $f(x) \le_{L} f(y)$; in fact $\leq_L$ is locally universal, i.e. for every pair $a<_{L} b$ and every c.e. pre ordering 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 $\{x: a \leq_{L} x \leq_{L} b\}$. Also $\leq_L$ is uniformly dense, i.e. there exists a computable function $f$ such that for every $a,b$ if $a<_{L} b$ then $a<_{L} f(a,b) <_{L} b$, and if $a\equiv_{L} a'$ and $b \equiv_{L} b'$ then $f(a,b)\equiv_{L} f(a',b')$. Some consequences and applications of these results are discussed: in particular for $n \ge 1$ the c.e. pre-ordering relation on $Σ_{n}$ 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 the c.e. pre-ordering relation of provable implication of Heyting Arithmetic is locally universal and uniformly dense.
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