Compatibility and accessibility: lattice representations for semantics of non-classical and modal logics

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. The third topological representation is a variant of that of Craig, Havier, and Priestley. We then add a second relation of accessibility interacting with compatibility in order to represent lattices with a multiplicative unary operation. The resulting representations yield an approach to semantics for modal logics on non-classical bases, motivated by a recent application of modal orthologic to natural language semantics.