natural and restricted priestley duality for ternary algebras and their cousins

up to term equivalence, there are three ways to assign a nonempty set c of constants to the three-element kleene lattice, leading to ternary algebras (c = {0, d, 1}), kleene algebras (c = {0, 1}), and don’t know algebras (c = {d}). our focus is on ternary algebras. we derive a strong, optimal natural duality and the restricted priestley duality for ternary algebras and give axiomatisations of the dual categories. we apply these dualities in tandem to give straightforward and transparent proofs of some known results for ternary algebras. we also discuss, and in some cases prove, the corresponding dualities for kleene lattices, kleene algebras and don’t know algebras.