کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
418578 | 681691 | 2011 | 12 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: Valuations and closure operators on finite lattices Valuations and closure operators on finite lattices](/preview/png/418578.png)
Let LL be a lattice. A function f:L→Rf:L→R (usually called evaluation) is submodular if f(x∧y)+f(x∨y)≤f(x)+f(y)f(x∧y)+f(x∨y)≤f(x)+f(y), supermodular if f(x∧y)+f(x∨y)≥f(x)+f(y)f(x∧y)+f(x∨y)≥f(x)+f(y), and modular if it is both submodular and supermodular. Modular functions on a finite lattice form a finite dimensional vector space. For finite distributive lattices, we compute this (modular) dimension. This turns out to be another characterization of distributivity (Theorem 3.9). We also present a correspondence between isotone submodular evaluations and closure operators on finite lattices (Theorem 5.5). This interplay between closure operators and evaluations should be understood as building a bridge between qualitative and quantitative data analysis.
Journal: Discrete Applied Mathematics - Volume 159, Issue 10, 28 June 2011, Pages 990–1001