有限集上偏序关系的符号计算

Symbolic Computation of Partial Orders on the Finite Set

不同于序理论和拓扑理论中关于偏序关系和T_(0)拓扑的研究思路,文章给出一种通过解有限域F_(2)上多项式方程组求有限集[n]={1,2,...,n}上所有偏序关系和T_(0)拓扑的方法,并通过实例说明了方程组零点和偏序以及T_(0)拓扑的对应关系.运用Grobner基理论,得到一种求有限集[n]上偏序个数和T_(0)拓扑个数的符号计算方法,并给出Maple程序.

Different from the known methods of order theory and topology theory on partial order and T_(0)topology,this paper presents a method to find all partial orders as well as T_(0)topologies on the finite set [n]={1,2,…,n} through solving a polynomial set over the finite field F_(2),and illustrates the correspondence between zeros of the polynomial set and partial orders as well as T_(0)topologies by examples.Based on Grobner bases theory,a symbolic computation method for computing the number of partial orders and the number of T_(0)topologies on [n] is obtained.Some examples are given to illustrate our method using Maple.