On Generalization of Definitional Equivalence to Languages with Non-Disjoint Signatures
On Generalization of Definitional Equivalence to Languages with Non-Disjoint Signatures
复制标题
关于具有非不相交签名的语言的定义等价的推广
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
G. Sz'ekely
中科院分区：
文献类型：
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作者：
Koen Lefever;G. Sz'ekely
为了简单起见，大多数文献只将定义等价的概念引入具有不相交签名的语言。在最近的一篇论文中，Barrett和Halvorson介绍了一个简单的推广到具有非不相交签名的语言，他们表明他们的推广并不等同于一般的互译性。本文证明了它们的推广不是传递的，因此不是等价关系。然后，我们介绍了Andreka和Nemeti的推广，作为不相交签名的语言的许多等价公式之一。我们证明了Andreka-Nemeti推广是包含Barrett-Halvorson推广的最小等价关系，并且即使对于具有非不相交签名的语言，它也等价于互译性。最后，我们研究了当我们将它们推广到具有非不相交签名的理论时，定义等价的定义仍然是等价的。
For simplicity, most of the literature introduces the concept of definitional equivalence only to languages with disjoint signatures. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to languages with non-disjoint signatures and they show that their generalization is not equivalent to intertranslatability in general. In this paper,we show that their generalization is not transitive and hence it is not an equivalence relation. Then we introduce the Andreka and Nemeti generalization as one of the many equivalent formulations for languages with disjoint signatures. We show that the Andreka-Nemeti generalization is the smallest equivalence relation containing the Barrett–Halvorson generalization and it is equivalent to intertranslatability even for languages with non-disjoint signatures. Finally,we investigate which definitions for definitional equivalences remain equivalent when we generalize them for theories with non-disjoint signatures.