Varieties of distributive lattices with unary operations. I. (English) Zbl 0907.06008

The paper sheds some new light on Cornish’s work on monoids acting on distributive lattices. More precisely, Priestley duality is developed for finitely generated varieties of the form \(\text{HSP}(P)\) – and more generally for classes of algebras of the form \(\text{ISP}(P)\) – where \(P\) is a finite distributive lattice with unary operators \(f_\mu\) which are assumed to be either an endomorphism or a dual endomorphism of the underlying lattice. As in Cornish’s work, the main assumption (easily fulfilled under many circumstances) is that the operators \(f_\mu\) are indexed by the points of the Priestley dual \(N\) of \(P\) and give rise to a semigroup action of \(N\) on itself. The main difference is that the author considers \(N\) as a weak ordered \(\pm\) semigroup, a concept that generalizes in an interesting manner Cornish’s \(\pm\) monoids.
This generalized point of view enables to consider specific varieties of distributive lattices in their own right, rather than as subvarieties of bigger ones. Many examples are given such as de Morgan, Kleene, MS and double MS algebras as well as varieties of Ockham algebras and others.
The author also fits her paper into the theory of natural dualities. In particular, conditions on \(N\) are given so that a naturally associated variety \({\mathcal A}^N\) of distributive lattices with operators has essentially the same natural and Priestley dualities. In this case, free algebras and coproducts are easily characterized.


06D05 Structure and representation theory of distributive lattices
06B20 Varieties of lattices