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

DOI:
https://doi.org/10.1090/S0002-9947-96-01601-7

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 of finitary algebraic relations yields a duality on a class of algebras , those subsets of which yield optimal dualities are characterised. Further, the manner in which the relations in are constructed from those in is revealed in the important special case that 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 . Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties 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

Email:
B.Davey@latrobe.edu.au

**H. A. Priestley**

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

Email:
hap@maths.ox.ac.uk

DOI:
https://doi.org/10.1090/S0002-9947-96-01601-7

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