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Optimal natural dualities: the structure of failsets

Gouveia, M. J.; (previously Saramago, M. J.); Priestley, H. A.

Internat. J. Algebra and Computation, 12 (3) (2002), 407-436

B.A.~Davey and H.A.~Priestley have investigated the optimality of dualities on a quasivariety $\a=\ISP (\m}$, where $\m$ is a finite algebra. Relative to a given set $\Omega$ of relations yielding a duality, they characterized the optimal dualities as the dualities determined by the transversals of a certain family of subsets of $\Omega$. However the structure of these subsets---known as globally minimal failsets---remained to be understood. This paper gives a complete description of the globally minimal failsets which do not contain partial endomorphisms, and an algorithmic method to determine them. These results are applied, by way of illustration, to the variety of de Morgan algebras and to two further varieties, one of them an Ockham algebra variety and the other a variety of Heyting algebras. All the globally minimal failsets are determined in each case.