具有等式的量词逻辑的有限化问题的一种解决方法
A solution to the finitizability problem for quantifier logics with equality
摘要：
我们考虑（ωω，\circ）的可数富子半群;每个这样的半群$T$给出一个簇CPEA_T，该簇可由取在具有等式的ω维圆柱-多进代数的可数子签名中的方程的有限模式公理化，其中置换限于T中的映射.证明了对任意的T，A有\in CPEA_T当且仅当A可表示为ω-元关系的具体集代数.签名中的运算被集合理论解释为像在多进相等集代数中一样，但是这样的运算相对于不一定不相交的笛卡儿空间的并集。这是一种保护语义的形式。我们证明了CPEA_T是正则的和原子正则的。通过对T附加一个条件，证明了CPEA_T中的原子代数是完全可表示的，并且CPEA_T具有超合并性质。如果T是丰富的和{ \it表示}，它表明CPEA_T是项定义等价于一个可公理化的Sahlqvist簇。这样的半群是存在的。如果我们不坚持量词的完全交换性，这可以看作是代数逻辑中一阶逻辑{ \it with equality}的中心有限性问题的一个解决方案。有限维的情况下，接近从观点的一阶逻辑片段的保护和集团保护（相对化）的语义使用多个变量。积极和消极的结果。
英文摘要：
We consider countable so-called rich subsemigroups of (ωω,\circ); each such semigroup $T$ gives a variety CPEA_T that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω-dimensional cylindric-polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, A\in CPEA_T iff A is representable as a concrete set algebra of ω-ary relations. The operations in the signature are set-theoretically interpreted like in polyadic equality set algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEA_T is canonical and atom-canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEA_T are completely representable and that CPEA_T has the super amalgamation property. If T is rich and {\it finitely represented}, it is shown that CPEA_T is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic {\it with equality} if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
