超拟阵的独立集公理、基公理和圈公理  

Independent set axioms, base axioms and circuit axioms of supermatroids

作　　者：李小南[1] 刘三阳[1] 易黄建[2] 

机构地区：[1]西安电子科技大学数学与统计学院,西安710071 [2]西北大学信息学院,西安710069

出　　处：《中国科学：数学》2016年第9期1351-1364,共14页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:61202178)资助项目

摘　　要：Dunstan等在1972年首先提出了超拟阵的概念,用以将定义拟阵的承载集合从有限集推广到偏序集.Barnabei等在1998年研究了另一种偏序集上的拟阵结构,即偏序集拟阵.由有限分配格和有限偏序集之间的对应关系可知,偏序集拟阵就是分配格上的超拟阵.本文研究超拟阵的公理系统,建立模格上的超拟阵的独立元公理,证明模格上超拟阵的中间基性质和基的交换性质并用这两个性质分别刻画了模超拟阵.最后指出了Barnabei等给出的分配超拟阵圈公理中的一个错误,重新提出并证明分配超拟阵的圈消去性质并建立了分配超拟阵的圈公理.作为圈消去性质的一个应用,本文证明了分配超拟阵中覆盖基的元素包含唯一的圈.Dunstan et al. first proposed the concept of supermatroids in 1972 by generalizing the underlying sets of matroids from finite sets to finite posers. Barnabei et al. introduced another matroidal structure on posets, i.e., poset matroids. By the one-to-one correspondence between finite distributive lattices and finite posers, poset matroids are just supermatroids on distributive lattices. This paper studies axiom systems of superinatroids. Independence axioms of modular supermatroids are proposed, middle base axiom and base exchange axiom of modular supermatroids are proved and modular supermatroids are characterized by these two properties. At last, we point out a mistake made by Barnabei et al. in their circuit axioms of distributive supermatroids and correct this mistake by proposing a new elimination property of circuit, then the circuit axiom of distributive supermatroids is established. As an application of the elimination property, it is proved that an element covering a base in a distributive supermatroid contains only one circtuit.

关 键 词：超拟阵 偏序集拟阵 模格 基公理 

分 类 号：O153.1[理学—数学]