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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**D.M. Clark and B.A. Davey,*The quest for strong dualities*, J. Austral. Math. Soc. (Series A)**58**(1995), 248--280. MR**96d:08010****2.**D.M. Clark and B.A. Davey,*Natural dualities for the working algebraist*, in preparation, Cambridge University Press.**3.**D.M. Clark and B.A. Davey,*When is a natural duality `good'?*, Algebra Universalis (to appear).**4.**B.A. Davey,*Duality theory on ten dollars a day*, Algebras and Orders, (I.G. Rosenberg and G. Sabidussi, eds.), NATO Advanced Study Institute Series, Series C, Vol. 389, Kluwer Academic Publishers, 1993, pp. 71--111. MR**94m:08001****5.**B.A. Davey, M. Haviar and H.A. Priestley,*The syntax and semantics of entailment in duality theory*, J. Symbolic Logic (to appear).**6.**B.A. Davey and H.A. Priestley,*Generalized piggyback dualities and applications to Ockham algebras*, Houston Math. J.**13**(1987), 151--197. MR**89f:06021****7.**B.A. Davey and H.A. Priestley,*Introduction to lattices and order*, Cambridge University Press, 1990. MR**91h:06001****8.**B.A. Davey and H.A. Priestley,*Partition-induced natural dualities for varieties of pseudocomplemented distributive lattices*, Discrete Math.**113**(1993), 41--58. MR**94i:06011****9.**B.A. Davey and H.A. Priestley,*Optimal natural dualities*, Trans. Amer. Math. Soc.**338**(1993), 655--677. MR**93j:06011****10.**B.A. Davey and H.A. Priestley,*Optimal natural dualities III: a miscellany of examples*, in preparation.**11.**B.A. Davey and H. Werner,*Dualities and equivalences for varieties of algebras*, Contributions to lattice theory (Szeged, 1980), (A.P. Huhn and E.T. Schmidt, eds.), Colloq. Math. Soc. János Bolyai, Vol. 33, North--Holland, Amsterdam, 1983, pp. 101--275. MR**85c:08012****12.**B.A. Davey and H. Werner,*Piggyback dualities*, Lectures in Universal Algebra (Szeged, 1983), (L. Szabó and A. Szendrei, eds.), Coll. Math. Soc. János Bolyai, Vol. 43, North--Holland, Amsterdam, vol. 43, 1986, pp. 61--83. MR**87i:08001****13.**B.A. Davey and H. Werner,*Piggyback-dualitäten*, Bull. Austral. Math. Soc.**32**(1985), 1--32. MR**87d:08002****14.**B. Jónsson,*Algebras whose congruence lattices are distributive*, Math. Scand.**21**(1967), 110--121. MR**38:5689****15.**A.F. Pixley,*Semi-categorical algebras II*, Math. Z.**85**(1964), 169-184. MR**29:5771****16.**H.A.Priestley,*Natural dualities*, Lattice theory and its applications---a volume in honor of Garrett Birkhoff's 80th Birthday, (K.A. Baker and R. Wille, eds.), Heldermann Verlag, 1995.**17.**H.A. Priestley and M.P. Ward,*A multi-purpose backtracking algorithm*, J. Symbolic Computation**18**(1994), 1--40. CMP**1995:3**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
08B99,
06D15,
06D05,
18A40

Retrieve articles in all journals with MSC (1991): 08B99, 06D15, 06D05, 18A40

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