有限偏序集上的剩余
Residuation in finite posets
摘要：
当研究一个基于偏序集而不是格的代数逻辑时，就有一个自然的问题，即如何引入处处定义的连接蕴涵，并满足与连接蕴涵的（左）伴随性。我们已经为基于正交模偏序集的量子力学逻辑或基于所谓的效应代数的量子效应逻辑研究了这个问题，效应代数只是部分的，不需要是格序的。为此，我们引入了算子剩余，其中蕴涵和合取的值不需要是基础偏序集的元素，而只需要是偏序集的某些子集，但这种方法可以推广到满足更一般条件的偏序集。如果这些偏序集是偶有限的，我们可以集中在相应子集的极大元或极小元上，并且所述算子的公式可以得到本质上的简化。这是在本文中所示的，其中所有的定理是由相应的例子解释。
英文摘要：
When an algebraic logic based on a poset instead of a lattice is investigated then there is a natural problem how to introduce the connective implication to be everywhere defined and satisfying (left) adjointness with the connective conjunction. We have already studied this problem for the logic of quantum mechanics which is based on an orthomodular poset or the logic of quantum effects based on a so-called effect algebra which is only partial and need not be lattice-ordered. For this, we introduced the so-called operator residuation where the values of implication and conjunction need not be elements of the underlying poset, but only certain subsets of it. However, this approach can be generalized for posets satisfying more general conditions. If these posets are even finite, we can focus on maximal or minimal elements of the corresponding subsets and the formulas for the mentioned operators can be essentially simplified. This is shown in the present paper where all theorems are explained by corresponding examples.
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