A FULL DUALITY THAT CANNOT BE UPGRADED TO A STRONG DUALITY

Abstract. We solve a variant of the Full Versus Strong Problem of natural duality theory, by giving an example of a three-element algebra that is fully dualisable but not strongly dualisable. This example also shows that there is no general “Strong Duality Compactness Theorem”. A full duality is a special kind of dual equivalence between a pair of categories A = ISP(M) and X = IScP+(M ∼), where M is an algebra and M ∼ is a compatible topological structure (called an alter ego of M). Examples include Pontryagin duality for abelian groups [14] (1934), Stone duality for Boolean algebras [16] (1936) and Priestley duality for distributive lattices [15] (1970). Until 2006, every known example of a full duality satisfied the additional prop-erty that the alter ego M ∼ was injective in X (called a strong duality). In general, a full duality must be strong if the base algebra M is injective in A [10, F.7]. There are also particular kinds of finite base algebras for which it is known that a full duality must be strong: semilattices, abelian groups, bounded distributive lattices and relative Stone algebras [7]. The first example of a full but not strong duality was found in 2006 by Clark, Davey and Willard [4], based on a four-element quasi-primal algebra. This full duality can nevertheless be upgraded to a strong duality, and so they asked the question: Is every fully dualisable algebra strongly dualisable? That is, if a finite algebra has an alter ego with which it yields a full duality, must it also have an alter ego with which it yields a strong duality