广义正交模偏序集的逻辑和代数性质
Logical and algebraic properties of generalized orthomodular posets
摘要：
广义正交模偏序集是D. Fazio，A. Ledda和本论文第一作者的工作，以建立一个有用的工具，研究量子力学的逻辑。他们研究了这些偏序集的结构特性。本文研究了这些偏序集的逻辑和代数性质。特别是，我们调查的条件下，他们可以转换成运营商剩余结构。此外，我们研究他们的表示代数（directoids）与无处不在的定义操作。我们证明同余性质的代数类分配给广义正交模偏序集，特别是，这一类的一个子品种由一个简单的身份。最后，与正交模偏序集的Dedekind-MacNeille完备化不一定是正交模格的事实相反，我们证明了广义正交模偏序集的强版本的Dedekind-MacNeille完备化几乎是正交模格。
英文摘要：
Generalized orthomodular posets were introduced recently by D. Fazio, A. Ledda and the first author of the present paper in order to establish a useful tool for studying the logic of quantum mechanics. They investigated structural properties of these posets. In the present paper we study logical and algebraic properties of these posets. In particular, we investigate conditions under which they can be converted into operator residuated structures. Further, we study their representation by means of algebras (directoids) with everywhere defined operations. We prove congruence properties for the class of algebras assigned to generalized orthomodular posets and, in particular, for a subvariety of this class determined by a simple identity. Finally, in contrast to the fact that the Dedekind-MacNeille completion of an orthomodular poset need not be an orthomodular lattice we show that the Dedekind-MacNeille completion of a stronger version of a generalized orthomodular poset is nearly an orthomodular lattice.
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