  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.
