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Optimal natural dualities. II: General theory

Authors: B. A. Davey and H. A. Priestley
Journal: Trans. Amer. Math. Soc. 348 (1996), 3673-3711
MSC (1991): Primary 08B99, 06D15, 06D05, 18A40
MathSciNet review: 1348858
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Abstract: 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.

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Additional Information

B. A. Davey
Affiliation: Department of Mathematics, La Trobe University, Bundoora, Victoria 3083, Australia

H. A. Priestley
Affiliation: Mathematical Institute, 24/29 St. Giles, Oxford OX1 3LB, England

Keywords: Natural duality, optimal duality
Received by editor(s): August 7, 1994
Received by editor(s) in revised form: August 29, 1995
Article copyright: © Copyright 1996 American Mathematical Society