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


[1] Balbes, R.; Dwinger, P., Distributive lattices, (1974), University of Missouri Press Columbia, MO · Zbl 0321.06012
[2] Berman, J., Notes on equational classes of algebras, (1974), University of Illinois Chicago, Mimeographed
[3] Burris, S.; Sankappanavar, H.P., A course in universal algebra, () · Zbl 0478.08001
[4] Cignoli, R., Boolean multiplicative closures. I & II, Proc. Japan acad., 42, 1168-1174, (1966) · Zbl 0149.25703
[5] Halmos, P.R.; Halmos, P.R., Algebraic logic, I. monadic Boolean algebras, Compositio math., Compositio math., 12, 217-249, (1955), Chelsea, New York · Zbl 0087.24505
[6] Halmos, P.R., Algebraic logic, (1962), Chelsea, New York · Zbl 0101.01101
[7] Janowitz, M.F., Quantifiers and orthomodular lattices, Pacific J. math, 13, 1241-1249, (1963) · Zbl 0144.25303
[8] Janowitz, M.F., Quantifier theory on quasi-orthomodular lattices, Illinois J. math., 9, 660-676, (1965) · Zbl 0146.01702
[9] Jónsson, B., Algebras whose congruence lattices are distributive, Math. scand., 21, 110-121, (1967) · Zbl 0167.28401
[10] Monk, D., On equational classes of algebraic versions of logic, I, Math. scand., 27, 53-71, (1970) · Zbl 0208.01202
[11] Monteiro, A., Normalidad de las álgebras de Heyting monádicas, Actas de las X jornadas de la unión matemática Argentina, bahia blanca, Notas de Lógica matemática no. 2, 50-51, (1974), Instituto de Matemática, Universidad Nacional del Sur Bahía Blaca, A French translation is published as
[12] Monteiro, A., Algebras monádicas, Atas do segundo colóquio brasileiro de matemática, Notas de Lógica matemátika no.7, 33-52, (1974), Instituto de Matimática, Universidad Nacional del Sur Bahía Blanca, A French translation was a published as
[13] Monteiro, A., Construction des algébres de lukasiewicz trivalentes dans LES algèbres de Boole monadiques I, Math. japon., 12, 1-23, (1967) · Zbl 0165.30903
[14] Monteiro, A.; Varsavsky, O., Algebras de Heyting monádicas, Actas de las X jornadas de la unión matemática Argentina, bahía blanca, Notas de logica mathemática no. 1, 52-62, (1974), Instituto de Matemática, Universidad Nacional del Sur Bahía Blanca, A French translation is published as
[15] Monteiro, L., Algébres de Boole monadiques libres, Algebra universalis, 8, 374-380, (1978) · Zbl 0381.03048
[16] Nachbin, L., Une propriété caractéristique des algèbres booléines, Protugal. math., 6, 115-118, (1947) · Zbl 0034.16603
[17] Priestley, H.A., Representation of distributive lattices by means of ordered stone spacs, Bull. London math. soc., 2, 186-190, (1970) · Zbl 0201.01802
[18] Priestley, H.A., Ordered topological spaces and the representation of distributive lattices, Proc. London math. soc., 4, 3, 507-530, (1972) · Zbl 0323.06011
[19] Priestly, H.A., Ordered sets and duality for distributive lattices, (), 39-60
[20] Varsavsky, O., Quantifiers and equivalence relations, Rev. matem. cuyana, Notas de Lógica matemática no 3. instituto de matemática, 2, 29-51, (1974), Universidad Nacional del Sur Bahía Blanca, Reproduced in · Zbl 0322.02054
[21] Vrancken-Mawet, L., The lattice of R-subalgebras of a bounded distributive lattice, Comment. math. univ. carolin., 25, 1-17, (1984) · Zbl 0542.06004
[22] Wallman, H., Lattices and topological spaces, Ann. of math., 39, 112-126, (1936) · JFM 64.0603.01
[23] Wright, F.B., Convergence of quantifiers and martingales, Illinois J. math., 6, 296-307, (1962) · Zbl 0112.09602
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