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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 \(\omega+1\)).

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