Natural dualities for varieties ofn-valued łukasiewicz algebras

Natural dualities are developed for varieties ofn-valued Łukasiewicz algebras with and without negation. These dualities are based on hom-functors, and parallel Stone duality for Boolean algebras. A translation is described which relates the natural dualities to the corresponding restricted Priestley dualities. This enables a unified approach to free algebras to be presented, whence R. Cignoli's characterisations of the finitely generated free algebras are elucidated and new descriptions of arbitrary free algebras obtained. Finally it is shown how dualities for subvarieties encode equational bases. An alternative approach is thereby provided to results of M.E. Adams and R. Cignoli.

[1]  Brian A. Davey.  Duality Theory on Ten Dollars a Day , 1993 .

[2]  Brian A. Davey,et al.  Partition-induced natural dualities for varieties of pseudo- complemented distributive lattices , 1993, Discret. Math..

[3]  B. Davey,et al.  The quest for strong dualities , 1995, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[4]  W. H. Cornish.  Antimorphic action : Categories of algebraic structures with involutions or anti-endomorphisms , 1986 .

[5]  Brian A. Davey,et al.  Exponentiation and Duality , 1982 .

[6]  Roberto Cignoli,et al.  Representation of Łukasiewicz and Post algebras by continuous functions , 1972 .

[7]  H. Priestley,et al.  Equational bases for varieties of Ockham algebras , 1994 .

[8]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[9]  George Georgescu,et al.  Lukasiewicz-Moisil algebras , 2012, Annals of discrete mathematics.

[10]  Hilary A. Priestley,et al.  Representation of Distributive Lattices by means of ordered Stone Spaces , 1970 .

[11]  M. E. Adams,et al.  A Note on the Axiomatization of Equational Classes of n-Valued Lukasiewicz Algebras , 1990, Notre Dame J. Formal Log..

[12]  Hilary A. Priestley,et al.  Ordered Sets and Duality for Distributive Lattices , 1984 .

[13]  Brian A. Davey,et al.  OPTIMAL NATURAL DUALITIES , 1993 .

[14]  H. Priestley.  The determination of subvarieties of certain congruence-distributive varieties , 1994 .

[15]  Alexandru Filipoiu.  Representation of Lukasiewicz algebras by means of ordered stone spaces , 1980, Discret. Math..