In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Plo\v{s}\v{c}ica. The standard representations of complete ortholattices and complete perfect Heyting algebras drop out as special cases of the first representation, while the second covers arbitrary complete lattices, as well as complete lattices equipped with a negation we call a protocomplementation. The third topological representation is a variant of that of Craig, Haviar, and Priestley. We then extend each of the three representations to lattices with a multiplicative unary modality; the representing structures, like so-called graph-based frames, add a second relation of accessibility interacting with compatibility. The three representations generalize possibility semantics for classical modal logics to non-classical modal logics, motivated by a recent application of modal orthologic to natural language semantics.
翻译:在本文中,我们通过一套符合Plo\v{s ⁇ v{c}ica传统兼容性的二元关系来研究三种拉特点的表达方式。 完整的正方形和完整的完美海代代数的标准表达方式作为第一个表达方式的特殊情况而退出, 而第二个表达方式则包括任意完整的拉特点, 以及配有我们所谓的否定的完整的拉特点。 第三个表层代表方式是克雷格、 哈维尔和皮斯利的变体。 我们随后将三种代表方式中的每一种扩展至具有多复制性单一模式的拉特克; 代表结构, 如所谓的图形框架, 增加了与兼容性互动的第二个关系。 三种表达方式将古典模式逻辑的语义概括为非古典模式逻辑的语义, 其动机是最近对自然语言的语义应用模式或图解学。