How Finite is a Three-Element Unary Algebra?

A full duality gives us a natural dual equivalence between a quasi-variety generated by a finite algebra and a class of structured topological spaces. There are many well-known examples of full dualities: Stone’s duality for Boolean algebras [12], Priestley’s duality for distributive lattices [11], and the Hofmann, Mislove, Stralka duality for semilattices [6]. A finite algebra M is called fully dualizable if it is possible to set up a full duality for the quasi-variety ISP(M). Given the naturalness of the definition of a full duality, it is perhaps surprising that the property of full dualizability is so difficult to work with. The subtlety of full dualizability is illuminated by the work done by Hyndman andWillard [8] to prove that a particular three-element unary algebra is not fully dualizable. There is a property that is stronger than full dualizability but easier to work with. Strong dualizability is more well behaved than full dualizability, and it is often easier to establish. Indeed, most full dualities are established by setting up a strong duality. It is not presently known whether every full duality is strong.