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Quantifiers on distributive lattices. (English) Zbl 0753.06012
The author studies (bounded) distributive lattices equipped with (the non-Boolean analogue of) a quantifier in the sense of P. R. Halmos [Compos. Math. 12, 217–249 (1956; Zbl 0087.24505)], that is a closure operator $$\nabla$$ which preserves finite joins (including 0) and satisfies the identity $$\nabla(a\land \nabla b)=\nabla a\land \nabla b$$. He shows that such operators on a given lattice $$L$$ correspond to equivalence relations on the Priestley space of $$L$$ satisfying suitable conditions. He also considers the variety of signature $$(2,2,1,0,0)$$ whose members are bounded distributive lattices equipped with a quantifier: he determines the finite subdirectly irreducible algebras in this variety (all of which have the “simple” quantifier which maps everything except 0 to 1) and its lattice of subvarieties (which turns out to be a chain of type $$\omega+1$$).

##### MSC:
 06D99 Distributive lattices 03G15 Cylindric and polyadic algebras; relation algebras 08B15 Lattices of varieties 08B26 Subdirect products and subdirect irreducibility
Zbl 0087.24505
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