{"title":"阿基米德向量网格截断的结构定理","authors":"Karim Boulabiar","doi":"10.1007/s00012-024-00858-4","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>X</i> be an Archimedean vector lattice and <span>\\(X_{+}\\)</span> denote the positive cone of <i>X</i>. A unary operation <span>\\(\\varpi \\)</span> on <span>\\(X_{+}\\)</span> is called a truncation on <i>X</i> if </p><div><div><span>$$\\begin{aligned} x\\wedge \\varpi \\left( y\\right) =\\varpi \\left( x\\right) \\wedge y\\quad \\text {for all }x,y\\in X_{+}. \\end{aligned}$$</span></div></div><p>Let <span>\\(X^{u}\\)</span> denote the universal completion of <i>X</i> with a distinguished weak element <span>\\(e>0.\\)</span> It is shown that a unary operation <span>\\(\\varpi \\)</span> on <span>\\(X_{+}\\)</span> is a truncation on <i>X</i> if and only if there exists an element <span>\\(u\\in X^{u}\\)</span> and a component <i>p</i> of <i>e</i> such that </p><div><div><span>$$\\begin{aligned} p\\wedge u=0\\quad \\text {and}\\quad \\varpi \\left( x\\right) =px+u\\wedge x\\ \\text {for all }x\\in X_{+}. \\end{aligned}$$</span></div></div><p>Here, <i>px</i> is the product of <i>p</i> and <i>x</i> with respect to the unique lattice-ordered multiplication in <span>\\(X^{u}\\)</span> having <i>e</i> as identity. As an example of illustration, if <span>\\(\\varpi \\)</span> is a truncation on some <span>\\(L_{p}\\left( {\\mu } \\right) \\)</span>-space then there exists a measurable set <i>A</i> and a function <span>\\(u\\in L_{0}\\left( {\\mu } \\right) \\)</span> vanishing on <i>A</i> such that <span>\\(\\varpi \\left( x\\right) =1_{A}x+u\\wedge x\\)</span> for all <span>\\(x\\in L_{p}\\left( {\\mu } \\right) .\\)</span></p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A structure theorem for truncations on an Archimedean vector lattice\",\"authors\":\"Karim Boulabiar\",\"doi\":\"10.1007/s00012-024-00858-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>X</i> be an Archimedean vector lattice and <span>\\\\(X_{+}\\\\)</span> denote the positive cone of <i>X</i>. A unary operation <span>\\\\(\\\\varpi \\\\)</span> on <span>\\\\(X_{+}\\\\)</span> is called a truncation on <i>X</i> if </p><div><div><span>$$\\\\begin{aligned} x\\\\wedge \\\\varpi \\\\left( y\\\\right) =\\\\varpi \\\\left( x\\\\right) \\\\wedge y\\\\quad \\\\text {for all }x,y\\\\in X_{+}. \\\\end{aligned}$$</span></div></div><p>Let <span>\\\\(X^{u}\\\\)</span> denote the universal completion of <i>X</i> with a distinguished weak element <span>\\\\(e>0.\\\\)</span> It is shown that a unary operation <span>\\\\(\\\\varpi \\\\)</span> on <span>\\\\(X_{+}\\\\)</span> is a truncation on <i>X</i> if and only if there exists an element <span>\\\\(u\\\\in X^{u}\\\\)</span> and a component <i>p</i> of <i>e</i> such that </p><div><div><span>$$\\\\begin{aligned} p\\\\wedge u=0\\\\quad \\\\text {and}\\\\quad \\\\varpi \\\\left( x\\\\right) =px+u\\\\wedge x\\\\ \\\\text {for all }x\\\\in X_{+}. \\\\end{aligned}$$</span></div></div><p>Here, <i>px</i> is the product of <i>p</i> and <i>x</i> with respect to the unique lattice-ordered multiplication in <span>\\\\(X^{u}\\\\)</span> having <i>e</i> as identity. As an example of illustration, if <span>\\\\(\\\\varpi \\\\)</span> is a truncation on some <span>\\\\(L_{p}\\\\left( {\\\\mu } \\\\right) \\\\)</span>-space then there exists a measurable set <i>A</i> and a function <span>\\\\(u\\\\in L_{0}\\\\left( {\\\\mu } \\\\right) \\\\)</span> vanishing on <i>A</i> such that <span>\\\\(\\\\varpi \\\\left( x\\\\right) =1_{A}x+u\\\\wedge x\\\\)</span> for all <span>\\\\(x\\\\in L_{p}\\\\left( {\\\\mu } \\\\right) .\\\\)</span></p></div>\",\"PeriodicalId\":50827,\"journal\":{\"name\":\"Algebra Universalis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-06-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebra Universalis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00012-024-00858-4\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra Universalis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00012-024-00858-4","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
让 X 是一个阿基米德向量网格,\(X_{+}\) 表示 X 的正锥。如果 $$\begin{aligned} x\wedge \varpi \left( y\right) =\varpi \left( x\right) \wedge y\quad \text {for all }x,y\in X_{+}, 那么在 \(X_{+}\) 上的一元操作 \(\varpi \) 称为 X 上的截断。\end{aligned}$Let \(X^{u}\) denote the universal completion of X with a distinguished weak element \(e>0.\当且仅当存在一个元素 \(u\in X^{u}\) 和一个 e 的分量 p,使得 $$$(X^{u}\) 上的一元运算 \(varpi \) 是 X 上的截断时,那么它就是 X 上的截断。分量 p,使得 $$begin{aligned} p\wedge u=0\quad \text {and}\quad \varpi \left( x\right) =px+u\wedge x\text {for all }x\in X_{+}.\end{aligned}$$这里,px 是 p 与 x 的乘积,与 \(X^{u}\)中以 e 为特征的唯一格子有序乘法有关。举例说明如果 \(\varpi \) 是某个 \(L_{p}\left( {\mu } \right) \)-空间上的一个截断空间,那么存在一个可测集合 A 和一个在 A 上消失的函数 (u\in L_{0}\left( {\mu } \right) \),使得 \(\varpi \left( x\right) =1_{A}x+u\wedge x\) for all \(xin L_{p}\left( {\mu } \right) .\)
A structure theorem for truncations on an Archimedean vector lattice
Let X be an Archimedean vector lattice and \(X_{+}\) denote the positive cone of X. A unary operation \(\varpi \) on \(X_{+}\) is called a truncation on X if
Let \(X^{u}\) denote the universal completion of X with a distinguished weak element \(e>0.\) It is shown that a unary operation \(\varpi \) on \(X_{+}\) is a truncation on X if and only if there exists an element \(u\in X^{u}\) and a component p of e such that
Here, px is the product of p and x with respect to the unique lattice-ordered multiplication in \(X^{u}\) having e as identity. As an example of illustration, if \(\varpi \) is a truncation on some \(L_{p}\left( {\mu } \right) \)-space then there exists a measurable set A and a function \(u\in L_{0}\left( {\mu } \right) \) vanishing on A such that \(\varpi \left( x\right) =1_{A}x+u\wedge x\) for all \(x\in L_{p}\left( {\mu } \right) .\)
期刊介绍:
Algebra Universalis publishes papers in universal algebra, lattice theory, and related fields. In a pragmatic way, one could define the areas of interest of the journal as the union of the areas of interest of the members of the Editorial Board. In addition to research papers, we are also interested in publishing high quality survey articles.