Natural and restricted Priestley duality for ternary algebras and their cousins

Document Type : Research Paper


1 Department of Mathematics, La Trobe University, Victoria 3086, Australia.

2 Department of Mathematics, La Trobe University, Victoria 3086, Australia


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.


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