具有伪补截的有限偏序集的蕴涵
Implication in finite posets with pseudocomplemented sections
摘要：
相对伪补格可以作为直觉主义逻辑的代数语义。为了将相对伪补的概念推广到非分配格，第一作者引入了所谓的分段伪补格，即顶元素为1的格，其中对于每个元素y，区间[y，1]，即所谓的分段，是伪补的。我们将这个概念推广到了具有顶元素的偏序集。我们的目标是证明这样的偏序集可以被认为是某种更一般的直觉主义逻辑的代数语义，只要引入一个蕴涵，如本文所示。我们证明了这样一个蕴涵的一些性质。这个蕴涵是“不清晰的”，因为给定元素的值不需要是唯一的元素，但可以是所讨论的偏序集的子集。另一方面，所有这些值都尽可能高。我们证明了这种蕴涵甚至决定了偏序集，并且如果以“不尖锐”的方式引入一个新的算子\odot，则这种结构形成一个“不尖锐”的剩余偏序集。
英文摘要：
It is well-known that relatively pseudocomplemented lattices can serve as an algebraic semantics of intuitionistic logic. To extend the concept of relative pseudocomplementation to non-distributive lattices, the first author introduced so-called sectionally pseudocomplemented lattices, i.e. lattices with top element 1 where for every element y the interval [y,1], the so called section, is pseudocomplemented. We extend this concept to posets with top element. Our goal is to show that such a poset can be considered as an algebraic semantics for a certain kind of more general intuitionistic logic provided an implication is introduced as shown in the paper. We prove some properties of such an implication. This implication is "unsharp" in the sense that the value for given entries need not be a unique element, but may be a subset of the poset in question. On the other hand, all of these values are as high as possible. We show that this implication even determines the poset and if a new operator \odot is introduced in an "unsharp" way, such structure forms an "unsharply" residuated poset.
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