## The order dimension of multinomial lattices.(English)Zbl 0795.06004

Summary: Using the notion of Ferrers dimension of incidence structures, the order dimension of multinomial lattices (i.e. lattices of ‘multi-permutations’) is determined. In particular, it is shown that the lattice of all permutations on an $$n$$-element set has dimension $$n-1$$.

### MSC:

 06B23 Complete lattices, completions 05A05 Permutations, words, matrices 06A07 Combinatorics of partially ordered sets
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### References:

 [1] M. K. Bennett and G. Birkhoff (1991) Two families of Newman lattices, Preprint, to appear inAlgebra Universalis. · Zbl 0810.06006 [2] A. Bouchet (1971) Etude combinatoire des ordonnés finis: Applications, Thèse de Doctorat d’Etat, Univ. Sci. et Méd., Grenoble. [3] O. Cogis (1980) La dimension Ferrers des graphes orientés, Thèse, Université Pierre et Marie Curie, Paris. [4] M. H. A. Newman (1942) On theories with a combinatorial definition of equivalence,Annals of Math. 43, 223-243. · Zbl 0060.12501 [5] K. Reuter (1989) On the dimension of the Cartesian product of relations and orders,Order 6, 277-293. · Zbl 0699.06004 [6] R. Wille (1982) Restructuring lattice theory: An approach based on hierarchies of concepts, inOrdered Sets (ed. I. Rival), D. Reidel, Dordrecht, 445-470. [7] R. Wille (1985) Tensorial decompositions of concept lattices,Order 2, 81-95. · Zbl 0583.06007 [8] R. Wille (1989) Lattices in data analysis: How to draw them with a computer, inAlgorithms and Order (ed. I. Rival), KAP, Dordrecht, 33-58. · Zbl 1261.68140 [9] T. Yanagimoto and M. Okamoto (1969) Partial orderings of permutations and monotonicity of a rank correlation statistic,Ann. Inst. Statist. Math. 21, 489-506. · Zbl 0208.44704
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