{"title":"Unbounded Order Convergence in Ordered Vector Spaces","authors":"Masoumeh Ebrahimzadeh, Kazem Haghnejad Azar","doi":"10.1155/2024/9960246","DOIUrl":null,"url":null,"abstract":"We consider an ordered vector space <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 10.0819 8.68572\" width=\"10.0819pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg>.</span> We define the net <span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 33.301 12.5794\" width=\"33.301pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,4.511,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,11.713,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,17.527,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,25.67,0)\"></path></g></svg><span></span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"36.8831838 -9.28833 10.171 12.5794\" width=\"10.171pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,36.933,0)\"><use xlink:href=\"#g113-89\"></use></g></svg></span> to be unbounded order convergent to <svg height=\"6.1673pt\" style=\"vertical-align:-0.2063904pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.39387 6.1673\" width=\"7.39387pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-121\"></use></g></svg> (denoted as <span><svg height=\"17.6182pt\" style=\"vertical-align:-3.291101pt\" version=\"1.1\" viewbox=\"-0.0498162 -14.3271 45.956 17.6182\" width=\"45.956pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-121\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,7.202,3.132)\"><use xlink:href=\"#g50-223\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,21.021,-8.782)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,25.853,-8.782)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,16.648,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,22.424,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,38.535,0)\"><use xlink:href=\"#g113-121\"></use></g></svg>).</span> This means that for every <span><svg height=\"12.0653pt\" style=\"vertical-align:-3.4294pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 17.503 12.0653\" width=\"17.503pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,9.872,0)\"></path></g></svg><span></span><svg height=\"12.0653pt\" style=\"vertical-align:-3.4294pt\" version=\"1.1\" viewbox=\"21.085183800000003 -8.6359 18.025 12.0653\" width=\"18.025pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,21.135,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,32.296,0)\"></path></g></svg><span></span><span><svg height=\"12.0653pt\" style=\"vertical-align:-3.4294pt\" version=\"1.1\" viewbox=\"42.7421838 -8.6359 10.185 12.0653\" width=\"10.185pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,42.792,0)\"><use xlink:href=\"#g113-89\"></use></g></svg>,</span></span> there exists a net <svg height=\"14.8173pt\" style=\"vertical-align:-5.52897pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 21.9833 14.8173\" width=\"21.9833pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-124\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.511,0)\"><use xlink:href=\"#g113-122\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,11.453,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,17.313,0)\"><use xlink:href=\"#g113-126\"></use></g></svg> (potentially over a different index set) such that <span><svg height=\"14.8695pt\" style=\"vertical-align:-5.52896pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.34054 22.986 14.8695\" width=\"22.986pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-122\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,6.942,3.132)\"><use xlink:href=\"#g50-224\"></use></g><g transform=\"matrix(.013,0,0,-0.013,16.434,0)\"></path></g></svg><span></span><span><svg height=\"14.8695pt\" style=\"vertical-align:-5.52896pt\" version=\"1.1\" viewbox=\"26.5681838 -9.34054 6.413 14.8695\" width=\"6.413pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,26.618,0)\"><use xlink:href=\"#g113-49\"></use></g></svg>,</span></span> and for every <span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 7.68094 12.7178\" width=\"7.68094pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg>,</span> there exists <svg height=\"9.25202pt\" style=\"vertical-align:-3.29111pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 12.1315 9.25202\" width=\"12.1315pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,7.059,3.132)\"></path></g></svg> such that <span><svg height=\"17.7871pt\" style=\"vertical-align:-5.5289pt\" version=\"1.1\" viewbox=\"-0.0498162 -12.2582 91.823 17.7871\" width=\"91.823pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-124\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.511,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,12.142,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,16.64,0)\"><use xlink:href=\"#g113-121\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,23.842,3.132)\"><use xlink:href=\"#g50-223\"></use></g><g transform=\"matrix(.013,0,0,-0.013,32.562,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,43.098,0)\"><use xlink:href=\"#g113-121\"></use></g><g transform=\"matrix(.013,0,0,-0.013,50.365,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,54.863,-5.741)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,60.404,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,65.547,0)\"><use xlink:href=\"#g113-122\"></use></g><g transform=\"matrix(.013,0,0,-0.013,73.076,0)\"><use xlink:href=\"#g113-126\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,77.587,-5.741)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,84.192,0)\"><use xlink:href=\"#g117-75\"></use></g></svg><span></span><svg height=\"17.7871pt\" style=\"vertical-align:-5.5289pt\" version=\"1.1\" viewbox=\"95.4051838 -12.2582 25.201 17.7871\" width=\"25.201pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,95.455,0)\"><use xlink:href=\"#g113-124\"></use></g><g transform=\"matrix(.013,0,0,-0.013,99.966,0)\"><use xlink:href=\"#g113-122\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,106.908,3.132)\"><use xlink:href=\"#g50-224\"></use></g><g transform=\"matrix(.013,0,0,-0.013,112.768,0)\"><use xlink:href=\"#g113-126\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,117.279,-5.741)\"><use xlink:href=\"#g50-109\"></use></g></svg></span> whenever <span><svg height=\"11.0658pt\" style=\"vertical-align:-3.29112pt\" version=\"1.1\" viewbox=\"-0.0498162 -7.77468 18.648 11.0658\" width=\"18.648pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-223\"></use></g><g transform=\"matrix(.013,0,0,-0.013,11.017,0)\"></path></g></svg><span></span><span><svg height=\"11.0658pt\" style=\"vertical-align:-3.29112pt\" version=\"1.1\" viewbox=\"22.230183800000002 -7.77468 12.187 11.0658\" width=\"12.187pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,22.28,0)\"><use xlink:href=\"#g113-223\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,29.339,3.132)\"><use xlink:href=\"#g50-49\"></use></g></svg>.</span></span> The emergence of a broader convergence, stemming from the recognition of more ordered vector spaces compared to lattice vector spaces, has prompted an expansion and broadening of discussions surrounding lattices to encompass additional spaces. We delve into studying the properties of this convergence and explore its relationships with other established order convergence. In every ordered vector space, we demonstrate that under certain conditions, every <span><svg height=\"6.1673pt\" style=\"vertical-align:-0.2063904pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 13.2267 6.1673\" width=\"13.2267pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,6.994,0)\"></path></g></svg>-</span>convergent net implies <span><svg height=\"6.1673pt\" style=\"vertical-align:-0.2063904pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 13.2267 6.1673\" width=\"13.2267pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-118\"></use></g><g transform=\"matrix(.013,0,0,-0.013,6.994,0)\"><use xlink:href=\"#g113-112\"></use></g></svg>-</span>Cauchy, and vice versa. Let <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 10.0819 8.68572\" width=\"10.0819pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-89\"></use></g></svg> be an order dense subspace of the directed ordered vector space <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 8.6074 8.68572\" width=\"8.6074pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg>.</span> If <span><svg height=\"10.5647pt\" style=\"vertical-align:-1.928801pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 17.142 10.5647\" width=\"17.142pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,9.511,0)\"><use xlink:href=\"#g117-75\"></use></g></svg><span></span><svg height=\"10.5647pt\" style=\"vertical-align:-1.928801pt\" version=\"1.1\" viewbox=\"20.724183800000002 -8.6359 8.655 10.5647\" width=\"8.655pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,20.774,0)\"><use xlink:href=\"#g113-90\"></use></g></svg></span> is a <span><svg height=\"6.1673pt\" style=\"vertical-align:-0.2063904pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 13.2267 6.1673\" width=\"13.2267pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-118\"></use></g><g transform=\"matrix(.013,0,0,-0.013,6.994,0)\"><use xlink:href=\"#g113-112\"></use></g></svg>-</span>band in <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 8.6074 8.68572\" width=\"8.6074pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-90\"></use></g></svg>,</span> then we establish that <span><svg height=\"10.5647pt\" style=\"vertical-align:-1.928801pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 16.416 10.5647\" width=\"16.416pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-75\"></use></g><g transform=\"matrix(.013,0,0,-0.013,8.785,0)\"></path></g></svg><span></span><svg height=\"10.5647pt\" style=\"vertical-align:-1.928801pt\" version=\"1.1\" viewbox=\"19.271183800000003 -8.6359 10.13 10.5647\" width=\"10.13pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,19.321,0)\"><use xlink:href=\"#g113-89\"></use></g></svg></span> is a <span><svg height=\"6.1673pt\" style=\"vertical-align:-0.2063904pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 13.2267 6.1673\" width=\"13.2267pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-118\"></use></g><g transform=\"matrix(.013,0,0,-0.013,6.994,0)\"><use xlink:href=\"#g113-112\"></use></g></svg>-</span>band in <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 10.0819 8.68572\" width=\"10.0819pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-89\"></use></g></svg>.</span>","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"11 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/9960246","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider an ordered vector space . We define the net to be unbounded order convergent to (denoted as ). This means that for every , there exists a net (potentially over a different index set) such that , and for every , there exists such that whenever . The emergence of a broader convergence, stemming from the recognition of more ordered vector spaces compared to lattice vector spaces, has prompted an expansion and broadening of discussions surrounding lattices to encompass additional spaces. We delve into studying the properties of this convergence and explore its relationships with other established order convergence. In every ordered vector space, we demonstrate that under certain conditions, every -convergent net implies -Cauchy, and vice versa. Let be an order dense subspace of the directed ordered vector space . If is a -band in , then we establish that is a -band in .
期刊介绍：
Journal of Mathematics is a broad scope journal that publishes original research articles as well as review articles on all aspects of both pure and applied mathematics. As well as original research, Journal of Mathematics also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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