| On the Maximal Disjoint Division in Riesz Spaces |
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| 作者姓名： | 熊洪允 苗新河 |
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| 作者单位： | School of Sciences，Tianjin University，School of Sciences，Tianjin University Tianjin 300072，China，Liu Hui Centre for Applied Mathematics，Tianjin 300072，China，Tianjin 300072，China，Liu Hui Centre for Applied Mathematics，Tianjin 300072，China |
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| 基金项目： | LiuHuiCentreforAppliedMathematicsNankaiUni versity&TianjinUniversity. |
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| 摘 要： | TheRieszspaces,consideredinthispaper ,areal waysArchimedean .LetLbeaRieszspace .Foru∈L ,theprincipalidealandtheprincipalbandgeneratedbyuaredenotedbyId(u)andB(u)respectively .ForA L ,theidealandthebandgeneratedbyAaredenotedbyId(A)andB(A)respectively .Aset {ui:i∈I}ofnonzeromutuallydisjointelementsofLiscalledamaximaldis jointsystem2 ] ifx⊥uiforalli∈Iimpliesx =0 (equiv alently ,theidealgeneratedby {ui:i∈I}isorderdenseinL) .AssumethatuiisaprojectionelementinL .ByPiwedenotetheorderp…
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On the Maximal Disjoint Division in Riesz Spaces |
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| XIONG Hong yun,MI AO Xin he.On the Maximal Disjoint Division in Riesz Spaces[J].Transactions of Tianjin University,2003,9(2). |
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| Authors: | XIONG Hong yun MI AO Xin he |
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| Abstract: | The authors point out a problem in the article of Ref. 1 ] ( Xiong Hongyun, Rong Ximin. Maximal disjoint systems in Riesz space and representationJ]. Acta Math Sinica,1998,41 (4) :763-766. )and revise it.Let E be an Archimedean Riesz space possessing a weak unit e and a maximal disjoint division { ei: i ∈ I | in which each ei is a projection element. Concerning the following statements: ( 1 ) There exists a completely regular Hausdorff space X such that E is Riesz isomorphic to C(X); (2) For every i ∈ I there exists a completely regular Hausdorff space Xi such that the band generated by ei is Riesz isomorphic to C (Xi). It is shown that (1) implies (2) ,and find some conditions for the inverse being true are found. Furthermore,if each Xi in (2)is a compact Hausdorff 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 laterally complete Riesz spaces are obtained. |
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| Keywords: | weak unit maximal disjoint division projection element Riesz homomorphism laterally complete |
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