Using coloured ordered sets to study finite-level full dualities

Abstract

We consider all the full dualities for the class of finite bounded distributive lattices that are based on the three-element chain 3. Under a natural quasi-order, these full dualities form a doubly algebraic lattice \({\mathcal{F}_{\underline{3}}}\). Using Priestley duality, we establish a correspondence between the elements of \({\mathcal{F}_{\underline{3}}}\) and special enriched ordered sets, which we call ‘coloured ordered sets’. We can then use combinatorial arguments to show that the lattice \({\mathcal{F}_{\underline{3}}}\) has cardinality \({2^{\aleph_{0}}}\) and is non-modular. This is the first investigation into the structure of an infinite lattice of finite-level full dualities.