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| name = "Matroids" | ||
| uuid = "f99c5f09-8689-4482-a891-4b988ec2fc3d" | ||
| authors = ["Ed Scheinerman <[email protected]>"] | ||
| version = "0.0.3" | ||
| version = "0.0.4" | ||
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| Graphs = "86223c79-3864-5bf0-83f7-82e725a168b6" | ||
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| {"documenter":{"julia_version":"1.11.3","generation_timestamp":"2025-02-06T19:09:30","documenter_version":"1.8.0"}} | ||
| {"documenter":{"julia_version":"1.11.3","generation_timestamp":"2025-02-06T20:19:04","documenter_version":"1.8.0"}} |
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| <!DOCTYPE html> | ||
| <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>What is a Matroid? · Matroids</title><meta name="title" content="What is a Matroid? · Matroids"/><meta property="og:title" content="What is a Matroid? · Matroids"/><meta property="twitter:title" content="What is a Matroid? · Matroids"/><meta name="description" content="Documentation for Matroids."/><meta property="og:description" content="Documentation for Matroids."/><meta property="twitter:description" content="Documentation for Matroids."/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-mocha.css" data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">Matroids</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Documentation</a></li><li><a class="tocitem" href="../design/">Design</a></li><li class="is-active"><a class="tocitem" href>What is a Matroid?</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>What is a Matroid?</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>What is a Matroid?</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/scheinerman/Matroids.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/scheinerman/Matroids.jl/blob/main/docs/src/math.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Core-Mathematical-Definitions"><a class="docs-heading-anchor" href="#Core-Mathematical-Definitions">Core Mathematical Definitions</a><a id="Core-Mathematical-Definitions-1"></a><a class="docs-heading-anchor-permalink" href="#Core-Mathematical-Definitions" title="Permalink"></a></h1><p>A <em>matroid</em> is a pair <span>$M=(S,\mathcal{I})$</span> where <span>$S$</span> is a set and <span>$\mathcal{I}$</span> is a set of subsets of <span>$S$</span> where:</p><ul><li>the emptyset <span>$\varnothing$</span> is an element of <span>$\mathcal{I}$</span> ,</li><li>if <span>$A \subseteq B \in \mathcal{I}$</span> then <span>$A \in \mathcal{I}$</span>, and</li><li>if <span>$A,B \in \mathcal{I}$</span> and <span>$|A| < |B|$</span>, then there is an <span>$x \in B - A$</span> such that <span>$A \cup\{x\} \in \mathcal{I}$</span>. </li></ul><p>The set <span>$S$</span> is called the <em>ground set</em> of <span>$M$</span> and the sets in <span>$\mathcal{I}$</span> are called <em>independent</em>. Refer to standard references for a more extensive introduction.</p><p>The <em>rank</em> of a matroid <span>$M=(S,\mathcal{I})$</span> is the size of a largest independent subset of <span>$S$</span>. The rank of <span>$M$</span> is the largest rank of a member of <span>$\mathcal{I}$</span>. An independent sets of maximum rank is called a <em>basis</em>. </p><p>See a matroid textbook or online resource for more detail. </p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../design/">« Design</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Thursday 6 February 2025 19:09">Thursday 6 February 2025</span>. Using Julia version 1.11.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> | ||
| <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>What is a Matroid? · Matroids</title><meta name="title" content="What is a Matroid? · Matroids"/><meta property="og:title" content="What is a Matroid? · Matroids"/><meta property="twitter:title" content="What is a Matroid? · Matroids"/><meta name="description" content="Documentation for Matroids."/><meta property="og:description" content="Documentation for Matroids."/><meta property="twitter:description" content="Documentation for Matroids."/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-mocha.css" data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">Matroids</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Documentation</a></li><li><a class="tocitem" href="../design/">Design</a></li><li class="is-active"><a class="tocitem" href>What is a Matroid?</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>What is a Matroid?</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>What is a Matroid?</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/scheinerman/Matroids.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/scheinerman/Matroids.jl/blob/main/docs/src/math.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Core-Mathematical-Definitions"><a class="docs-heading-anchor" href="#Core-Mathematical-Definitions">Core Mathematical Definitions</a><a id="Core-Mathematical-Definitions-1"></a><a class="docs-heading-anchor-permalink" href="#Core-Mathematical-Definitions" title="Permalink"></a></h1><p>A <em>matroid</em> is a pair <span>$M=(S,\mathcal{I})$</span> where <span>$S$</span> is a set and <span>$\mathcal{I}$</span> is a set of subsets of <span>$S$</span> where:</p><ul><li>the empty set <span>$\varnothing$</span> is an element of <span>$\mathcal{I}$</span> ,</li><li>if <span>$A \subseteq B \in \mathcal{I}$</span> then <span>$A \in \mathcal{I}$</span>, and</li><li>if <span>$A,B \in \mathcal{I}$</span> and <span>$|A| < |B|$</span>, then there is an <span>$x \in B - A$</span> such that <span>$A \cup\{x\} \in \mathcal{I}$</span>. </li></ul><p>The set <span>$S$</span> is called the <em>ground set</em> of <span>$M$</span> and the sets in <span>$\mathcal{I}$</span> are called <em>independent</em>. Refer to standard references for a more extensive introduction.</p><p>The <em>rank</em> of a matroid <span>$M=(S,\mathcal{I})$</span> is the size of a largest independent subset of <span>$S$</span>. The rank of <span>$M$</span> is the largest rank of a member of <span>$\mathcal{I}$</span>. An independent sets of maximum rank is called a <em>basis</em>. </p><p>See a matroid textbook or <a href="https://en.wikipedia.org/wiki/Matroid">online resource</a> for more detail. </p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../design/">« Design</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Thursday 6 February 2025 20:19">Thursday 6 February 2025</span>. Using Julia version 1.11.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
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