On Tamari lattices. (English) Zbl 0811.06005

The Tamari lattice \(T_ n\) is defined as the set of all binary bracketings on \(n+1\) symbols ordered by applying the associative rule only in one direction. Using a vector representation it is proved that this order defines a lattice. To determine the structure of Tamari lattices, the author uses methods of formal concept analysis. The Tamari lattices are bounded subdirectly irreducible lattices. A construction method for these lattices is given. Tamari lattices and their congruence lattices have the same number of elements.


06B05 Structure theory of lattices
06B10 Lattice ideals, congruence relations


Zbl 0778.06007
Full Text: DOI


[1] M.K. Bennett and G. Birkhoff, Two families of Newman lattices, Algebra Universalis, to appear. · Zbl 0810.06006
[2] Comtet, L., Advanced combinatorics, (1974), Reidel Dordrecht
[3] Day, A., Splitting lattices generate all lattices, Algebra universalis, 7, 163-170, (1977) · Zbl 0381.06010
[4] Day, A., Characterizations of finite lattices that are bounded-homomorphic images or sublattices of free lattices, Canad. J. math., 31, 69-78, (1979) · Zbl 0432.06007
[5] Geyer, W., Generalizing semidistributivity, 10, 77-92, (1993), Order · Zbl 0813.06007
[6] W. Geyer, The generalized doubling construction and formal concept analysis, Algebra Universalis, to appear. · Zbl 0829.06007
[7] Huang, S.; Tamari, D., Problems of associativity: a simple proof for the lattice property of systems ordered by a semi-associative law, J. combin. theory, 13, 7-13, (1972), Ser. A · Zbl 0248.06003
[8] Markowsky, G., Primes, irreducibles and extremal lattices, 9, 265-290, (1992), Order · Zbl 0778.06007
[9] Narayana, T.V., Sur LES Trellis formĂ© par LES partitions d’un entier, C.R. acad. sci., 240, 1188-1189, (1955) · Zbl 0064.12705
[10] Nation, J.B., Finite sublattices of a free lattice, Trans. amer. math. soc., 269, 311-337, (1982) · Zbl 0507.06007
[11] Simion, R.; Ullman, D., On the structure of the lattice of noncrossing partitions, Discrete math., 98, 193-206, (1991) · Zbl 0760.05004
[12] Tamari, D., The algebra of bracketings and their enumeration, Nieuw arch. wiskd. III. ser., 10, 131-146, (1962) · Zbl 0109.24502
[13] Urquhart, A., A topological representation theory for lattices, Algebra universalis, 8, 45-58, (1978) · Zbl 0382.06010
[14] Wille, R., Restructuring lattice theory: an approach based on hierarchies of concepts, (), 445-470
[15] Wille, R., Subdirect decomposition of concept lattices, Algebra universalis, 17, 275-287, (1983) · Zbl 0539.06006
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