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 $\mathcal {A} = \operatorname {\mathbb {I}\mathbb {S}\mathbb {P}}( \underline {M})$, 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 $\underline {M}$ 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 $\underline {M}$. Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties $\mathbf {B}_{n}$ of pseudocomplemented distributive lattices, the theory improves upon and illuminates previous results.