关于分配格上矩阵方程的一个注记

A Note on the Matrix Equations In a Distributive Lattice

设L是一个分配格,L的每个元素都有有限的并既约分解。在此条件下,本文给出了L上矩阵方程AX=B的全体解的求法,这大大减弱了文[1]中的条件。

Let L be a distributive lattice with the property that every element of L has a finite jion-decomposition into joint irreducible elements. Matrix equations of the form AX=B are considered in the lattice L, where A=(a_(ij))_(m×n), B=(b_1,…,b_m)~T, a_(ij), b_i∈L. Necessary and sufficient conditions for the solvability of the equations are obtained. In the solvable case, the entire solution set are determined. This paper weakens the conditions of paper [1]. In paper [1], we require that the lattice L is a Brouwerian lattic and every element of L has a finite jion—decomposition into joint irreducible elements, but in this paper we only require the last one.