The authors p oint out a problem in the article of Ref.(Xiong Hongyun,Rong Ximin.Maximal disjoint systems in Riesz space and representation.Acta Math Sinica ,1998,41(4):763-766.)and revise it.Let E be an Archimed ean ...The authors p oint out a problem in the article of Ref.(Xiong Hongyun,Rong Ximin.Maximal disjoint systems in Riesz space and representation.Acta Math Sinica ,1998,41(4):763-766.)and revise it.Let E be an Archimed ean Riesz space possessing a weak unit e and a maximal disjoint division{e i:i∈I} in which each e i is a proj ection element. Concerning the following statements:(1) There exists a completel y regular Hausdorff space X such that E is Riesz isomorphic to C(X);(2) For every i∈I there exi sts a completely regular Hausdorff space X i such that the band generated by e i is Riesz isomorphic to C(X i).It is shown that (1) implies (2),and find some conditions for the inverse bein g true are found.Furthermore,if each X i in (2) is a compact Ha usdorff space, a necessary and sufficient condition is established under which E can be represented as a C(X) for some compact Hausdorff space X.As corollaries, corresponding results for late rally complete Riesz spaces are obtained.展开更多
文摘The authors p oint out a problem in the article of Ref.(Xiong Hongyun,Rong Ximin.Maximal disjoint systems in Riesz space and representation.Acta Math Sinica ,1998,41(4):763-766.)and revise it.Let E be an Archimed ean Riesz space possessing a weak unit e and a maximal disjoint division{e i:i∈I} in which each e i is a proj ection element. Concerning the following statements:(1) There exists a completel y regular Hausdorff space X such that E is Riesz isomorphic to C(X);(2) For every i∈I there exi sts a completely regular Hausdorff space X i such that the band generated by e i is Riesz isomorphic to C(X i).It is shown that (1) implies (2),and find some conditions for the inverse bein g true are found.Furthermore,if each X i in (2) is a compact Ha usdorff space, a necessary and sufficient condition is established under which E can be represented as a C(X) for some compact Hausdorff space X.As corollaries, corresponding results for late rally complete Riesz spaces are obtained.
