Optimal Natural Dualities. Ii: General Theory

A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set R of finitary algebraic relations yields a duality on a class of algebras A = ISP(M), those subsets R′ of R which yield optimal dualities are characterised. Further, the manner in which the relations in R are constructed from those in R′ is revealed in the important special case that M generates a congruence-distributive variety and is such that each of its subalgebras is subdirectly irreducible. These results are obtained by studying a certain algebraic closure operator, called entailment, definable on any set of algebraic relations on M . Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties Bn of pseudocomplemented distributive lattices, the theory improves upon and illuminates previous results.