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：本文沿袭Ploščica的研究传统，研究了通过具有兼容性二元关系的集合所实现的三种格表示。完备正交格和完备完美海廷代数的标准表示可视为第一种表示的特例，而第二种表示涵盖了任意完备格，以及配备了我们称为原始补余化（protocomplementation）的否定运算的完备格。第三种拓扑表示是Craig、Haviar和Priestley表示的变体。随后，我们将这三种表示分别推广到具有乘法一元模态算子的格上；其表示结构在所谓的图式框架（graph-based frames）中，增加了与兼容性相互作用的第二种可及性关系。这三种表示将经典模态逻辑的可能性语义推广至非经典模态逻辑，这一推广源于模态正规模态逻辑在自然语言语义学中的最新应用。