Natural dualities for varieties of distributive lattices with a quantifier.(English)Zbl 0799.06023

Rauszer, Cecylia (ed.), Algebraic methods in logic and in computer science. Papers of the XXXVIII semester on algebraic methods in logic and their computer science applications held in Warsaw (Poland) between September 15 and December 15, 1991. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 28, 291-310 (1993).
R. Cignoli [Discrete Math. 96, No. 3, 183-197 (1991; Zbl 0753.06012)] derived a dual category equivalence between $$Q$$-distributive lattices (= distributive lattices with a quantifier) and a category of $$Q$$-spaces, and gave a construction of the $$Q$$-space dual to the $$Q$$- algebra freely generated by a bounded distribution lattice.
For every variety of $$Q$$-distributive lattices, the author presents a duality based on hom-functors into a schizophrenic object, and, in quite a different way, via the natural dualities describes free algebras in every variety $$D_{pq}$$ of $$Q$$-distributive lattices needing neither the identities which determine $$D_{pq}$$ (so far unknown beyond $$D_{01}$$) nor a subcategory of Cignoli’s category dual to $$D_{pq}$$.
For the entire collection see [Zbl 0777.00048].
Reviewer: L.Esakia (Tbilisi)

MSC:

 06D99 Distributive lattices 08B20 Free algebras 06B20 Varieties of lattices 03G99 Algebraic logic

Zbl 0753.06012