推廣施-董的組合固定點定理至有限分配格

Abstract

施-董的組合不動點定理證明，如果從n維超立方體到自身的函數滿足了每個在超立方體的元素其布爾雅可比矩陣的特徵值是零，那麼該函數有唯一的固定點。該定理等價對偶敘述具有生物學意義。我們的目標是推廣施-董定理到所有的有限分配格。我們的證明方法是基於施-董的“集體影響法”以及G.伯克霍夫的有限分配格表現定理。

Shih-Dong's combinational fixed point theorem asserts that if a map from the n-dimensional hypercube into itself satisfies that all the Boolean eigenvalues of the Boolean Jacobian matrix are zero for each element in the hypercube, then it has a unique fixed point. Its equivalent contrapositive form has biological implications. Our goal is to provide an extension of Shih-Dong's theorem into all finite distributive lattices. Our method of proof is based on Shih-Dong's “collective effect method” as well as G. Birkhoff's representation theorem for finite distributive lattices.