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<!DOCTYPE html> <html lang="en"> <head> <title>BooleanGB -- Groebner Bases for Ideals in Boolean Polynomial Quotient Ring</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="Groebner Bases for Ideals in Boolean Polynomial Quotient Ring" href="index.html">BooleanGB</a> :: <a title="Groebner Bases for Ideals in Boolean Polynomial Quotient Ring" href="index.html">BooleanGB</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_gb__Boolean_lp__Ideal_rp.html">next</a> | previous | <a href="_gb__Boolean_lp__Ideal_rp.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>BooleanGB -- Groebner Bases for Ideals in Boolean Polynomial Quotient Ring</h1> <div> <h2>Description</h2> <div> <p>BooleanGB is a package to compute Groebner Bases in lexicographic order for polynomial ideals in the quotient ring $\mathbb F_2[x_1, \ldots, x_n]/J$, where J is the ideal generated by field polynomials $x_i^2 - x_i $ for $i \in \{ 1, \ldots, n\}$. The algorithm is implemented bitwise rather than symbolically, which reduces the computational complexity.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : n = 3;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : R = ZZ/2[vars(0)..vars(n-1)];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : J = apply( gens R, x -> x^2 + x);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : QR = R/J;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : I = ideal(a+b,b); o5 : Ideal of QR</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : gbBoolean I o6 = ideal (b, a) o6 : Ideal of QR</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : gens gb I o7 = | b a | 1 2 o7 : Matrix QR <-- QR</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> <div> <p>BooleanGB always assumes that the ideal is in the Boolean quotient ring, i.e., $\mathbb F_2[x_1, \ldots, x_n] / <x_1^2-x_1, \ldots, x_n^2-x_n >$, regardless of the ring in which the ideal was generated. Thus, any ideal in the base ring is promoted to the quotient ring automatically, even if the quotient ring has not been defined.</p> </div> </div> <div> <div> <div> <h2>Authors</h2> <ul> <li><a href="http://www.math.vt.edu/people/fhinkel/">Franziska Hinkelmann</a><span> <<a href="mailto:fhinkel%40vt.edu">fhinkel@vt.edu</a>></span></li> <li>Mike Stillman</li> <li>Elizabeth Arnold</li> </ul> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>1.0</b> of BooleanGB, released <b>May 9, 2011</b>.</p> </div> <div> <h2>Citation</h2> <p>If you have used this package in your research, please cite it as follows:</p> <table class="examples"> <tr> <td> <pre><code class="language-bib">@misc{BooleanGBSource, title = {{BooleanGB: Groebner bases for ideals in Boolean polynomial quotient rings. Version~1.0}}, author = {Franziska Hinkelmann and Mike Stillman and Elizabeth Arnold}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Functions and commands <ul> <li><span><kbd>gbBoolean</kbd> -- see <span><a title="Compute Groebner Basis for Ideals in Boolean Polynomial Quotient Ring" href="_gb__Boolean_lp__Ideal_rp.html">gbBoolean(Ideal)</a> -- Compute Groebner Basis for Ideals in Boolean Polynomial Quotient Ring</span></span></li> </ul> </li> <li>Methods <ul> <li><span><a title="Compute Groebner Basis for Ideals in Boolean Polynomial Quotient Ring" href="_gb__Boolean_lp__Ideal_rp.html">gbBoolean(Ideal)</a> -- Compute Groebner Basis for Ideals in Boolean Polynomial Quotient Ring</span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Groebner Bases for Ideals in Boolean Polynomial Quotient Ring" href="index.html">BooleanGB</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">BooleanGB.m2</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">BooleanGB.m2:50:0</span>.</p> </div> </div> </div> </body> </html>