{"id":3026,"date":"2025-06-07T15:03:53","date_gmt":"2025-06-07T15:03:53","guid":{"rendered":"https:\/\/diznr.com\/?p=3026"},"modified":"2025-06-07T15:03:53","modified_gmt":"2025-06-07T15:03:53","slug":"day-03part-09-cseit-cross-structure-failure-case-of-greatest-lower-bound-and-least-bound-upper","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-03part-09-cseit-cross-structure-failure-case-of-greatest-lower-bound-and-least-bound-upper\/","title":{"rendered":"Day 03Part 09-CseIt- Cross structure failure case of greatest lower bound and least upper bound."},"content":{"rendered":"<p>Day 03Part 09-CseIt- Cross structure failure case of greatest lower bound and least upper bound.<\/p>\n<p>[fvplayer id=&#8221;222&#8243;]<\/p>\n<p class=\"\" data-start=\"0\" data-end=\"74\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><strong data-start=\"0\" data-end=\"116\" data-is-last-node=\"\" data-is-only-node=\"\">Day 03 Part 09 \u2013 CSE\/IT: Cross Structure Failure Cases of Greatest Lower Bound (GLB) and Least Upper Bound (LUB)<\/strong><\/span><\/p>\n<hr class=\"\" data-start=\"76\" data-end=\"79\" \/>\n<h3 class=\"\" data-start=\"81\" data-end=\"127\">\ud83d\udd0d <strong data-start=\"88\" data-end=\"127\">Understanding GLB and LUB in Posets<\/strong><\/h3>\n<p class=\"\" data-start=\"129\" data-end=\"210\">In a <strong data-start=\"134\" data-end=\"167\">partially ordered set (poset)<\/strong>, for any two elements <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span>:<\/p>\n<ul data-start=\"212\" data-end=\"391\">\n<li class=\"\" data-start=\"212\" data-end=\"283\">\n<p class=\"\" data-start=\"214\" data-end=\"283\"><strong data-start=\"214\" data-end=\"244\">Greatest Lower Bound (GLB)<\/strong>: <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">The largest element that is less than or equal to both <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span>.<\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"284\" data-end=\"391\">\n<p class=\"\" data-start=\"286\" data-end=\"391\"><strong data-start=\"286\" data-end=\"313\">Least Upper Bound (LUB)<\/strong>: <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">The smallest element that is greater than or equal to both <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span>.<\/span><\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"393\" data-end=\"471\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">A <strong data-start=\"2\" data-end=\"13\">lattice<\/strong> is a poset where every pair of elements has both a GLB and a LUB.<\/span><\/p>\n<hr class=\"\" data-start=\"473\" data-end=\"476\" \/>\n<h3 class=\"\" data-start=\"478\" data-end=\"517\">\u274c <strong data-start=\"484\" data-end=\"517\">Cross Structure Failure Cases<\/strong><\/h3>\n<p class=\"\" data-start=\"519\" data-end=\"597\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Certain poset configurations, known as <strong data-start=\"39\" data-end=\"59\">cross structures<\/strong>, fail to be lattices because some pairs of elements lack a GLB or LUB.<\/span><span class=\"ms-1 inline-flex max-w-full items-center relative top-[-0.094rem] animate-[show_150ms_ease-in]\"><span class=\"relative start-0 bottom-0 flex h-full w-full items-center\"><span class=\"flex h-4 w-full items-center justify-between overflow-hidden\"><span class=\"max-w-full grow truncate overflow-hidden text-center\">Mathematics Stack Exchange<\/span><\/span><\/span><\/span><\/p>\n<h4 class=\"\" data-start=\"599\" data-end=\"630\"><strong data-start=\"604\" data-end=\"630\">Example 1: Missing LUB<\/strong><\/h4>\n<p class=\"\" data-start=\"632\" data-end=\"711\">Consider the poset with elements <span class=\"katex\"><span class=\"katex-mathml\">{a,b,c}\\{a, b, c\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span> and the following relations:<\/p>\n<ul data-start=\"713\" data-end=\"835\">\n<li class=\"\" data-start=\"713\" data-end=\"754\">\n<p class=\"\" data-start=\"715\" data-end=\"754\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> are incomparable.<\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"755\" data-end=\"835\">\n<p class=\"\" data-start=\"757\" data-end=\"835\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Both <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> are less than <span class=\"katex\"><span class=\"katex-mathml\">cc<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><\/span><\/span><\/span>.<\/span><\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"837\" data-end=\"855\"><strong data-start=\"837\" data-end=\"855\">Hasse Diagram:<\/strong><\/p>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary\">\n<div class=\"flex items-center text-token-text-secondary px-4 py-2 text-xs font-sans justify-between h-9 bg-token-sidebar-surface-primary dark:bg-token-main-surface-secondary select-none rounded-t-[5px]\">css<\/div>\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-sidebar-surface-primary text-token-text-secondary dark:bg-token-main-surface-secondary flex items-center rounded-sm px-2 font-sans text-xs\"><button class=\"flex gap-1 items-center select-none px-4 py-1\" aria-label=\"\u0915\u0949\u092a\u0940 \u0915\u0930\u0947\u0902\">\u0915\u0949\u092a\u0940 \u0915\u0930\u0947\u0902<\/button><span class=\"\" data-state=\"closed\"><button class=\"flex items-center gap-1 px-4 py-1 select-none\">\u092c\u0926\u0932\u0947\u0902<\/button><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre!\"><br \/>\n  c<br \/>\n \/ \\<br \/>\n<span class=\"hljs-selector-tag\">a<\/span>   <span class=\"hljs-selector-tag\">b<\/span><br \/>\n<\/code><\/div>\n<\/div>\n<p class=\"\" data-start=\"881\" data-end=\"920\">\n<p class=\"\" data-start=\"922\" data-end=\"935\"><strong data-start=\"922\" data-end=\"935\">Analysis:<\/strong><\/p>\n<ul data-start=\"937\" data-end=\"1123\">\n<li class=\"\" data-start=\"937\" data-end=\"1010\">\n<p class=\"\" data-start=\"939\" data-end=\"1010\"><strong data-start=\"939\" data-end=\"969\">GLB of <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span><\/strong>: <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Does not exist since <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> are incomparable and have no common lower bound.<\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"1011\" data-end=\"1123\">\n<p class=\"\" data-start=\"1013\" data-end=\"1123\"><strong data-start=\"1013\" data-end=\"1043\">LUB of <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span><\/strong>: <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Exists and is <span class=\"katex\"><span class=\"katex-mathml\">cc<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><\/span><\/span><\/span>.<\/span><\/p>\n<\/li>\n<\/ul>\n<h4 class=\"\" data-start=\"1125\" data-end=\"1156\"><strong data-start=\"1130\" data-end=\"1156\">Example 2: Missing GLB<\/strong><\/h4>\n<p class=\"\" data-start=\"1158\" data-end=\"1237\">Consider the poset with elements <span class=\"katex\"><span class=\"katex-mathml\">{a,b,d}\\{a, b, d\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span> and the following relations:<\/p>\n<ul data-start=\"1239\" data-end=\"1361\">\n<li class=\"\" data-start=\"1239\" data-end=\"1280\">\n<p class=\"\" data-start=\"1241\" data-end=\"1280\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> are incomparable.<\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"1281\" data-end=\"1361\">\n<p class=\"\" data-start=\"1283\" data-end=\"1361\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Both <span class=\"katex\"><span class=\"katex-mathml\">dd<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">d<\/span><\/span><\/span><\/span> is less than <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span>.<\/span><\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"1363\" data-end=\"1381\"><strong data-start=\"1363\" data-end=\"1381\">Hasse Diagram:<\/strong><\/p>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary\">\n<div class=\"flex items-center text-token-text-secondary px-4 py-2 text-xs font-sans justify-between h-9 bg-token-sidebar-surface-primary dark:bg-token-main-surface-secondary select-none rounded-t-[5px]\">css<\/div>\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-sidebar-surface-primary text-token-text-secondary dark:bg-token-main-surface-secondary flex items-center rounded-sm px-2 font-sans text-xs\"><button class=\"flex gap-1 items-center select-none px-4 py-1\" aria-label=\"\u0915\u0949\u092a\u0940 \u0915\u0930\u0947\u0902\">\u0915\u0949\u092a\u0940 \u0915\u0930\u0947\u0902<\/button><span class=\"\" data-state=\"closed\"><button class=\"flex items-center gap-1 px-4 py-1 select-none\">\u092c\u0926\u0932\u0947\u0902<\/button><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre!\"><br \/>\n<span class=\"hljs-selector-tag\">a<\/span>   <span class=\"hljs-selector-tag\">b<\/span><br \/>\n \\ \/<br \/>\n  d<br \/>\n<\/code><\/div>\n<\/div>\n<p class=\"\" data-start=\"1407\" data-end=\"1446\">\n<p class=\"\" data-start=\"1448\" data-end=\"1461\"><strong data-start=\"1448\" data-end=\"1461\">Analysis:<\/strong><\/p>\n<ul data-start=\"1463\" data-end=\"1649\">\n<li class=\"\" data-start=\"1463\" data-end=\"1536\">\n<p class=\"\" data-start=\"1465\" data-end=\"1536\"><strong data-start=\"1465\" data-end=\"1495\">GLB of <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span><\/strong>: <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Exists and is <span class=\"katex\"><span class=\"katex-mathml\">dd<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">d<\/span><\/span><\/span><\/span>.<\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"1537\" data-end=\"1649\">\n<p class=\"\" data-start=\"1539\" data-end=\"1649\"><strong data-start=\"1539\" data-end=\"1569\">LUB of <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span><\/strong>: <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Does not exist since <span class=\"katex\"><span class=\"katex-mathml\">aa<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> are incomparable and have no common upper bound.<\/span><span class=\"ms-1 inline-flex max-w-full items-center relative top-[-0.094rem] animate-[show_150ms_ease-in]\"><span class=\"relative start-0 bottom-0 flex h-full w-full items-center\"><span class=\"flex h-4 w-full items-center justify-between overflow-hidden\"><span class=\"max-w-full grow truncate overflow-hidden text-center\">focs.computer.org<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<hr class=\"\" data-start=\"1651\" data-end=\"1654\" \/>\n<h3 class=\"\" data-start=\"1656\" data-end=\"1680\">\ud83e\udde0 <strong data-start=\"1663\" data-end=\"1680\">Key Takeaways<\/strong><\/h3>\n<ul data-start=\"1682\" data-end=\"1804\">\n<li class=\"\" data-start=\"1682\" data-end=\"1723\">\n<p class=\"\" data-start=\"1684\" data-end=\"1723\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">In cross structures, certain pairs of elements lack either a GLB or LUB, preventing the poset from being a lattice.<\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"1724\" data-end=\"1804\">\n<p class=\"\" data-start=\"1726\" data-end=\"1804\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Identifying these failure cases is crucial in discrete mathematics and computer science, especially in topics like compiler design and formal language theory.<\/span><\/p>\n<\/li>\n<\/ul>\n<hr class=\"\" data-start=\"1806\" data-end=\"1809\" \/>\n<p class=\"\" data-start=\"1811\" data-end=\"1900\">If you need further clarification or additional examples on this topic, feel free to ask!<\/p>\n<h3 data-start=\"1811\" data-end=\"1900\"><a href=\"https:\/\/www.math.uh.edu\/~dblecher\/C2seq.pdf\" target=\"_blank\" rel=\"noopener\">Day 03Part 09-CseIt- Cross structure failure case of greatest lower bound and least upper bound.<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Day 03Part 09-CseIt- Cross structure failure case of greatest lower bound and least upper bound. [fvplayer id=&#8221;222&#8243;] Day 03 Part 09 \u2013 CSE\/IT: Cross Structure Failure Cases of Greatest Lower Bound (GLB) and Least Upper Bound (LUB) \ud83d\udd0d Understanding GLB and LUB in Posets In a partially ordered set (poset), for any two elements aaa [&hellip;]<\/p>\n","protected":false},"author":71,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[76],"tags":[],"class_list":["post-3026","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3026","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/71"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=3026"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3026\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=3026"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=3026"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=3026"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}