Natural and restricted Priestley duality for ternary algebras and their cousins

Up to term equivalence, there are three ways to assign a nonemptyset C of constants to the three-element Kleene lattice, leading toternary algebras (C = {0, d, 1}), Kleene algebras (C = {0, 1}), and don’tknow algebras (C = {d}). Our focus is on ternary algebras. We derivea strong, optimal natural duality and the restricted Priestley duality forternary algebras and give axiomatisations of the dual categories. We applythese dualities in tandem to give straightforward and transparent proofsof some known results for ternary algebras. We also discuss, and in somecases prove, the corresponding dualities for Kleene lattices, Kleene algebrasand don’t know algebras.