Optimal natural dualities

Authors:
B. A. Davey and H. A. Priestley

Journal:
Trans. Amer. Math. Soc. **338** (1993), 655-677

MSC:
Primary 06D15

MathSciNet review:
1169079

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The authors showed previously that for each of the varieties of pseudocomplemented distributive lattices there exists a natural duality given by a set of binary algebraic relations, where denotes the number of partitions of . This paper improves this result by establishing that an optimal set of of these relations suffices. This is achieved by the use of "test algebras": it is shown that redundancy among the relations of a duality for a prevariety generated by a finite algebra may be decided by testing the duality on the relations, *qua* algebras.

**[1]**Brian A. Davey,*Topological duality for prevarieties of universal algebras*, Studies in foundations and combinatorics, Adv. in Math. Suppl. Stud., vol. 1, Academic Press, New York-London, 1978, pp. 61–99. MR**520555****[2]**Brian A. Davey,*Dualities for Stone algebras, double Stone algebras, and relative Stone algebras*, Colloq. Math.**46**(1982), no. 1, 1–14. MR**672356****[3]**B. A. Davey and H. A. Priestley,*Generalised piggyback dualities and applications to Ockham algebras*, Houston J. Math.**13**(1987), no. 2, 151–198. MR**904949****[4]**-,*Partition-induced natural dualities for varieites of pseudocomplemented distributive lattices*, Discrete Math. (to appear).**[5]**B. A. Davey and H. Werner,*Dualities and equivalences for varieties of algebras*, Contributions to lattice theory (Szeged, 1980) Colloq. Math. Soc. János Bolyai, vol. 33, North-Holland, Amsterdam, 1983, pp. 101–275. MR**724265****[6]**B. A. Davey and H. Werner,*Piggyback dualities*, Lectures in universal algebra (Szeged, 1983) Colloq. Math. Soc. János Bolyai, vol. 43, North-Holland, Amsterdam, 1986, pp. 61–83. MR**860256****[7]**-,*Piggyback-dualitäten*, Bull. Austral. Math. Soc.**32**(1985), 1-32.**[8]**George Grätzer,*Lattice theory. First concepts and distributive lattices*, W. H. Freeman and Co., San Francisco, Calif., 1971. MR**0321817****[9]**H. Peter Gumm,*Geometrical methods in congruence modular algebras*, Mem. Amer. Math. Soc.**45**(1983), no. 286, viii+79. MR**714648**, 10.1090/memo/0286**[10]**H. A. Priestley,*The construction of spaces dual to pseudocomplemented distributive lattices*, Quart. J. Math. Oxford Ser. (2)**26**(1975), no. 102, 215–228. MR**0392731****[11]**H. A. Priestley and M. P. Ward,*A multi-purpose backtracking algorithm*.**[12]**M. H. Stone,*The theory of representations for Boolean algebras*, Trans. Amer. Math. Soc.**40**(1936), no. 1, 37–111. MR**1501865**, 10.1090/S0002-9947-1936-1501865-8**[13]**M. P. Ward, D. Phil. thesis, University of Oxford, 1989.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
06D15

Retrieve articles in all journals with MSC: 06D15

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1993-1169079-7

Keywords:
Distributive -algebra,
natural duality,
piggyback duality,
optimal duality

Article copyright:
© Copyright 1993
American Mathematical Society