Tarski关系代数的保持定理
Preservation theorems for Tarski's relation algebra
AI总结：
本文研究Tarski二元关系代数各语义片段的有限生成性，证明同态安全及若干前向片段可有限生成，而保持函数与全函数片段不可由有限个受卫二阶运算表达。
摘要：
我们研究了Tarski二元关系代数的若干语义定义片段，包括保持函数的片段。我们探讨它们是否由有限个运算生成的问题，并沿此方向获得了若干肯定与否定结果。具体而言，同态安全片段是有限生成的（在有限结构和任意结构上均如此）。保持函数的片段不是有限生成的（事实上，它无法由任何有限个受卫二阶可定义的保持函数运算表达）。类似地，保持全函数的片段也不是有限生成的（事实上，它无法由任何有限个受卫二阶可定义的保持全函数运算表达）。相比之下，前向保持函数的片段可由复合、交、反定义域和优先并有限生成。类似地，前后向保持单射函数的片段可由复合、交、反定义域、逆以及一个“单射并”运算有限生成。
英文摘要：
We investigate a number of semantically defined fragments of Tarski's algebra of binary relations, including the function-preserving fragment. We address the question whether they are generated by a finite set of operations. We obtain several positive and negative results along these lines. Specifically, the homomorphism-safe fragment is finitely generated (both over finite and over arbitrary structures). The function-preserving fragment is not finitely generated (and, in fact, not expressible by any finite set of guarded second-order definable function-preserving operations). Similarly, the total-function-preserving fragment is not finitely generated (and, in fact, not expressible by any finite set of guarded second-order definable total-function-preserving operations). In contrast, the forward-looking function-preserving fragment is finitely generated by composition, intersection, antidomain, and preferential union. Similarly, the forward-and-backward-looking injective-function-preserving fragment is finitely generated by composition, intersection, antidomain, inverse, and an `injective union' operation.
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