Coordinatization of finite join-distributive lattices

作者：Czedli Gabor^*

来源：ALGEBRA UNIVERSALIS, 2014, 71(4): 385-404.

DOI：10.1007/s00012-014-0282-3

• semimodular lattice
• diamond-free lattice
• planar lattice
• Jordan-Holder permutation
• antimatroid
• convex geometry
• anti-exchange closure

Join-distributive lattices are finite, meet-semidistributive, and semimodular lattices. They are the same as lattices with unique irreducible decompositions, introduced by R.P. Dilworth in 1940, and many alternative definitions and equivalent concepts have been discovered or rediscovered since then. Let L be a join-distributive lattice of length n, and let k denote the width of the set of join-irreducible elements of L. We prove that there exist k - 1 permutations acting on {1, . . . , n} such that the elements of L are coordinatized by k-tuples over {0, . . . , n}, and the permutations determine which k-tuples are allowed. Since the concept of join-distributive lattices is equivalent to that of antimatroids and convex geometries, our result offers a coordinatization for these combinatorial structures.

• 出版日期2014-6