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Varieties of distributive lattices with unary operations. II. (English) Zbl 0912.06009

In his paper “Varieties of distributive lattices with unary operations. I” [J. Aust. Math. Soc., Ser. A 63, 165-207 (1997; Zbl 0907.06008)], the first author generalized and adapted Cornish’s work on monoids acting on distributive lattices to the setting of natural dualities. In the sequel, the whole machinery of generalized dualities is needed to obtain a natural duality for all subvarieties of a given finitely generated variety of distributive lattice ordered algebras. Free algebras and coproducts are then easily described. The whole theory is fully illustrated in the case of double MS-algebras and equational bases are given – through an algorithmic process – for each of the 21 meet-irreducible subvarieties of double MS-algebras. The latter results should be compared to those of Varlet and Vaz de Cavalho on the same subject.

MSC:

06D05 Structure and representation theory of distributive lattices
06B20 Varieties of lattices
08A60 Unary algebras
08B20 Free algebras
08B25 Products, amalgamated products, and other kinds of limits and colimits
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)

Citations:

Zbl 0907.06008
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