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Varieties of distributive lattices with unary operations I

  • H. A. Priestley (a1)

Abstract

A unified study is undertaken of finitely generated varieties HSP () of distributive lattices with unary operations, extending work of Cornish. The generating algebra () is assusmed to be of the form (P; ∧, ∨, 0, 1, {fμ}), where each fμ is an endomorphism or dual endomorphism of (P; ∧, ∨, 0, 1), and the Priestly dual of this lattice is an ordered semigroup N whose elements act by left multiplication to give the maps dual to the operations fμ. Duality theory is fully developed within this framework, into which fit many varieties arising in algebraic logic. Conditions on N are given for the natural and Priestley dualities for HSP () to be essentially the same, so that, inter alia, coproducts in HSP () are enriched D-coproducts.

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References

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[1]Adams, M. E. and Priestley, H. A., ‘Equational bases for varieties of Ockham algebras’, Algebra Universalis 32 (1994), 368397.
[2]Balbes, R. and Dwinger, Ph., Distributive lattices (University of Missouri Press, Columbia, 1974).
[3]Blyth, T. S., Noor, A. S. A. and Varlet, J. C., ‘Ockham algebras with de Morgan skeletons’, J. Algebra 117 (1988), 165178.
[4]Blyth, T. S., Noor, A. S. A. and Varlet, J. C., ‘Equational bases for subvarieties of double MS-algebras’, Glasgow Math. J. 31 (1989), 116.
[5]Blyth, T. S. and Varlet, J. C., ‘subdirectly irreducible double MS-algebras’, Proc. Roy. Soc. Edinburgh Sect. A 98 (1994), 241247.
[6]Blyth, T. S. and Varlet, J. C., Ockham algebras (Oxford University Press, Oxford, 1994).
[7]Burris, S. and Sankappanavar, H. P., A course in universal algebra (Springer, New York, 1981).
[8]Cignoli, R. and de Gallego, M. S., ‘The lattice structure of some Łukasiewicz algebras’, Algebra Universalis 13 (1981), 315328.
[9]Cignoli, R. and de Gallego, M. S., ‘Dualities for some De Morgan algebras with operators and Łukasiewicz algebras’, J. Austral. Math. Soc. (Series A) 34 (1983), 377393.
[10]Clark, D. M. and Davey, B. A., Natural dualities for the working algebraist, in preparation (Cambridge University Press).
[11]Cornish, W. H., ‘Monoids acting on distributive lattices’, manuscript (invited talk at the annual meeting of the Austral. Math. Soc., May 1977).
[12;]Cornish, W. H., Antimorphic action. Categories of algebraic structures with involutions or antiendomorphisms, R & E Res. Exp. Math., 12 (Heldermann Verlag, Berlin, 1986).
[13]Cornish, W. H. and Fowler, P. R., ‘Coproducts of Kleene algebras’, J. Austral. Math. Soc. (Series A) 27 (1979), 209220.
[14]Davey, B. A., ‘On the lattice of subvarieties’, Houston J. Math. 5 (1979), 183192.
[15]Davey, B. A., ‘Duality theory on ten dollars a day’, in: Algebras and orders (eds. Rosenberg, I. G. and Sabidussi, G.) NATO Advanced Study Institute Series, Series C, 389 (Kluwer, Dordrecht, 1993), 71111.
[16]Davey, B. A. and Priestley, H. A., ‘Lattices of homomorphisms’, J. Austral. Math. Soc. (Series A) 40 (1986), 364406.
[17]Davey, B. A. and Priestley, H. A., ‘Generalised piggyback dualities and applications to Ockham algebras’, Houston J. Math. 13 (1987), 151197.
[18]Davey, B. A. and Priestley, H. A., Introduction to lattices and order (Cambridge University Press, Cambridge, 1990).
[19]Davey, B. A. and Werner, H., ‘Dualities and equivalences for varieties of algebras’, in: Contributions to lattice theory (Szeged, 1980) (eds. Huhn, A. P. and Schmidt, E. T.) Colloq. Math. Soc. János Bolyai 33, (North-Holland, Amsterdam 1983), pp. 101275.
[20]Davey, B. A. and Werner, H., ‘Piggyback-dualitäten’, Bull. Austral. Math. Soc. 32 (1985), 132.
[21]Davey, B. A. and Werner, H., ‘Piggyback dualities’, in: Colloq. Math. Soc. Jhnos Bolyai 43 (1986), 6183.
[22]Goldberg, M. S., ‘Distributive Ockham algebras: free algebras and injectivity’, Bull. Austral. Math. Soc. 24 (1981), 161203.
[23]Guzman, F. and Squier, C. C., ‘Subdirectly irreducible and free Kleene-Stone algebras’, Algebra Universalis 31 (1994), 266273.
[24]Priestley, H. A., ‘Ordered sets and duality for distributive lattices’, in: Orders, description and roles (eds. Pouzet, M. and Richard, D.), Ann. Discrete Math. 23 (North-Holland, Amsterdam, 1984), pp. 3960.
[25]Priestley, H. A., ‘The determination of the lattices of subvarieties of certain congruence-distributive varieties’, Algebra Universalis 32 (1994), 4462.
[26]Priestley, H. A., ‘Natural dualities’, in: Lattice theory and its applications—a volume in honor of Garrett Birkhoff's 80th birthday (eds. Baker, K. A. and Wille, R.) (Heldermann Verlag, Berlin, 1995), pp. 185209.
[27]Priestley, H. A., ‘A note on equational bases’, Bull. Soc. Roy. Sci. Liège 64 (1995), 113118.
[28]Priestley, H. A., ‘Natural dualities for varieties of n-valued Lukasiewicz algebras’, Studia Logica 54, 333370.
[29]Priestley, H. A., ‘Dualities and identities I’, preprint.
[30]Priestley, H. A. and Santos, R., ‘Dualities and identities II: DMS-algebras’, preprint.
[31]Priestley, H. A. and Santos, R., ‘Varieties of distributive lattices with unary operations II’, Portugal Math., to appear.
[32]Santos, R., ‘Involutive Stone algebras and regular α De Morgan algebras’, manuscript.
[33]Santos, R., ‘Natural dualities for some subvarieties of DMS’, manuscript.
[34]Sankappanavar, H. P., ‘Linked double weak Stone algebras’, Z. Math. Logik Grundlag. Math. 35 (1989), 485494.
[35]Sequeira, M., ‘Double MS n-algebras and double K m.n-algebras’, Glasgow Math. J. 35 (1993), 189201.
[36]Urquhart, A., ‘Lattices with a dual homomorphic operation’, Studia Logica 38 (1979), 201209.
[37]Urquhart, A., ‘Lattices with a dual homomorphic operation II’, Studia Logica 40 (1981), 391404.
[38]Varlet, J. C., ‘MS-algebras: a survey’, in: Lattices, semigroups and universal algebra (Lisbon, 1988) (eds. Almeida, J., Bordalo, G. and Dwinger, Ph.) (Plenum Press, New York, 1990), pp. 299313.
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Journal of the Australian Mathematical Society
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  • EISSN: 1446-8107
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