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    <title>Binomials -- a package for binomial ideals</title>
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@article{BinomialsArticle,
  title = {{Decompositions of binomial ideals}},
  author = {Thomas Kahle},
  journal = {The Journal of Software for Algebra and Geometry: Macaulay2},
  volume = {4},
  year = {2012},
}
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            <h2>Exports</h2>
            <div class="exports">
              <ul>
                <li>Types                  <ul>
                    <li><span><a title="the class of all partial characters" href="___Partial__Character.html">PartialCharacter</a> -- the class of all partial characters</span></li>
                  </ul>
                </li>
                <li>Functions and commands                  <ul>
                    <li><span><a title="Binomial Cellular Decomposition" href="___B__C__D.html">BCD</a> -- Binomial Cellular Decomposition</span></li>
                    <li><span><a title="Associated primes of a binomial ideal" href="_binomial__Associated__Primes.html">binomialAssociatedPrimes</a> -- Associated primes of a binomial ideal</span></li>
                    <li><span><a title="Binomial Cellular Decomposition" href="_binomial__Cellular__Decomposition.html">binomialCellularDecomposition</a> -- Binomial Cellular Decomposition</span></li>
                    <li><span><a title="test for primary binomial ideals" href="_binomial__Is__Primary.html">binomialIsPrimary</a> -- test for primary binomial ideals</span></li>
                    <li><span><a title="test for primeness of a binomial ideal" href="_binomial__Is__Prime.html">binomialIsPrime</a> -- test for primeness of a binomial ideal</span></li>
                    <li><span><a title="minimal primes of a binomial Ideal" href="_binomial__Minimal__Primes.html">binomialMinimalPrimes</a> -- minimal primes of a binomial Ideal</span></li>
                    <li><span><a title="Binomial Primary Decomposition" href="_binomial__Primary__Decomposition.html">binomialPrimaryDecomposition</a> -- Binomial Primary Decomposition</span></li>
                    <li><span><a title="Radical of a binomial ideal" href="_binomial__Radical.html">binomialRadical</a> -- Radical of a binomial ideal</span></li>
                    <li><span><a title="solving zero-dimensional binomial Ideals" href="_binomial__Solve.html">binomialSolve</a> -- solving zero-dimensional binomial Ideals</span></li>
                    <li><span><a title="Binomial Unmixed Decomposition" href="_binomial__Unmixed__Decomposition.html">binomialUnmixedDecomposition</a> -- Binomial Unmixed Decomposition</span></li>
                    <li><span><a title="Binomial Primary Decomposition" href="___B__P__D.html">BPD</a> -- Binomial Primary Decomposition</span></li>
                    <li><span><a title="Binomial Unmixed Decomposition" href="___B__U__D.html">BUD</a> -- Binomial Unmixed Decomposition</span></li>
                    <li><span><a title="Associated primes of a cellular binomial ideal" href="_cellular__Binomial__Associated__Primes.html">cellularBinomialAssociatedPrimes</a> -- Associated primes of a cellular binomial ideal</span></li>
                    <li><span><a title="test for primaryness of a binomial ideal" href="_cellular__Binomial__Is__Primary.html">cellularBinomialIsPrimary</a> -- test for primaryness of a binomial ideal</span></li>
                    <li><span><a title="Primary decomposition of a cellular binomial ideal" href="_cellular__Binomial__Primary__Decomposition.html">cellularBinomialPrimaryDecomposition</a> -- Primary decomposition of a cellular binomial ideal</span></li>
                    <li><span><a title="Radical of a cellular binomial ideal" href="_cellular__Binomial__Radical.html">cellularBinomialRadical</a> -- Radical of a cellular binomial ideal</span></li>
                    <li><span><a title="Unmixed decomposition of a cellular binomial ideal" href="_cellular__Binomial__Unmixed__Decomposition.html">cellularBinomialUnmixedDecomposition</a> -- Unmixed decomposition of a cellular binomial ideal</span></li>
                    <li><span><a title="Extract inclusion minimal ideals from a list of ideals" href="_extract__Inclusion__Minimal__Ideals.html">extractInclusionMinimalIdeals</a> -- Extract inclusion minimal ideals from a list of ideals</span></li>
                    <li><span><a title="Generate a lattice ideal from a character." href="_ideal__From__Character.html">idealFromCharacter</a> -- Generate a lattice ideal from a character.</span></li>
                    <li><span><a title="testing for unital binomial ideals" href="_is__Binomial.html">isBinomial</a> -- testing for unital binomial ideals</span></li>
                    <li><span><kbd>isUnital</kbd> -- see <span><a title="testing for unital binomial ideals" href="_is__Binomial.html">isBinomial</a> -- testing for unital binomial ideals</span></span></li>
                    <li><span><a title="testing for cellular binomial ideals" href="_is__Cellular.html">isCellular</a> -- testing for cellular binomial ideals</span></li>
                    <li><span><a title="construct the ideal whose generators correspond to generators of an integer lattice" href="_lattice__Basis__Ideal.html">latticeBasisIdeal</a> -- construct the ideal whose generators correspond to generators of an integer lattice</span></li>
                    <li><span><a title="make a binomial from an exponent vector and a coefficient" href="_make__Binomial.html">makeBinomial</a> -- make a binomial from an exponent vector and a coefficient</span></li>
                    <li><span><a title="Computing the partial character of a cellular binomial ideal" href="_partial__Character.html">partialCharacter</a> -- Computing the partial character of a cellular binomial ideal</span></li>
                    <li><span><a title="Random Binomial Ideals" href="_random__Binomial__Ideal.html">randomBinomialIdeal</a> -- Random Binomial Ideals</span></li>
                  </ul>
                </li>
                <li>Methods                  <ul>
                    <li><span><kbd>binomialCellularDecomposition(Ideal)</kbd> -- see <span><a title="Binomial Cellular Decomposition" href="_binomial__Cellular__Decomposition.html">binomialCellularDecomposition</a> -- Binomial Cellular Decomposition</span></span></li>
                    <li><span><kbd>binomialIsPrime(Ideal)</kbd> -- see <span><a title="test for primeness of a binomial ideal" href="_binomial__Is__Prime.html">binomialIsPrime</a> -- test for primeness of a binomial ideal</span></span></li>
                    <li><span><kbd>binomialMinimalPrimes(Ideal)</kbd> -- see <span><a title="minimal primes of a binomial Ideal" href="_binomial__Minimal__Primes.html">binomialMinimalPrimes</a> -- minimal primes of a binomial Ideal</span></span></li>
                    <li><span><kbd>binomialPrimaryDecomposition(Ideal)</kbd> -- see <span><a title="Binomial Primary Decomposition" href="_binomial__Primary__Decomposition.html">binomialPrimaryDecomposition</a> -- Binomial Primary Decomposition</span></span></li>
                    <li><span><kbd>binomialUnmixedDecomposition(Ideal)</kbd> -- see <span><a title="Binomial Unmixed Decomposition" href="_binomial__Unmixed__Decomposition.html">binomialUnmixedDecomposition</a> -- Binomial Unmixed Decomposition</span></span></li>
                    <li><span><kbd>cellularBinomialAssociatedPrimes(Ideal)</kbd> -- see <span><a title="Associated primes of a cellular binomial ideal" href="_cellular__Binomial__Associated__Primes.html">cellularBinomialAssociatedPrimes</a> -- Associated primes of a cellular binomial ideal</span></span></li>
                    <li><span><kbd>cellularBinomialIsPrimary(Ideal)</kbd> -- see <span><a title="test for primaryness of a binomial ideal" href="_cellular__Binomial__Is__Primary.html">cellularBinomialIsPrimary</a> -- test for primaryness of a binomial ideal</span></span></li>
                    <li><span><kbd>cellularBinomialPrimaryDecomposition(Ideal)</kbd> -- see <span><a title="Primary decomposition of a cellular binomial ideal" href="_cellular__Binomial__Primary__Decomposition.html">cellularBinomialPrimaryDecomposition</a> -- Primary decomposition of a cellular binomial ideal</span></span></li>
                    <li><span><kbd>cellularBinomialRadical(Ideal)</kbd> -- see <span><a title="Radical of a cellular binomial ideal" href="_cellular__Binomial__Radical.html">cellularBinomialRadical</a> -- Radical of a cellular binomial ideal</span></span></li>
                    <li><span><kbd>cellularBinomialUnmixedDecomposition(Ideal)</kbd> -- see <span><a title="Unmixed decomposition of a cellular binomial ideal" href="_cellular__Binomial__Unmixed__Decomposition.html">cellularBinomialUnmixedDecomposition</a> -- Unmixed decomposition of a cellular binomial ideal</span></span></li>
                    <li><span><kbd>extractInclusionMinimalIdeals(List)</kbd> -- see <span><a title="Extract inclusion minimal ideals from a list of ideals" href="_extract__Inclusion__Minimal__Ideals.html">extractInclusionMinimalIdeals</a> -- Extract inclusion minimal ideals from a list of ideals</span></span></li>
                    <li><span><kbd>idealFromCharacter(Ring,PartialCharacter)</kbd> -- see <span><a title="Generate a lattice ideal from a character." href="_ideal__From__Character.html">idealFromCharacter</a> -- Generate a lattice ideal from a character.</span></span></li>
                    <li><span><kbd>isCellular(Ideal)</kbd> -- see <span><a title="testing for cellular binomial ideals" href="_is__Cellular.html">isCellular</a> -- testing for cellular binomial ideals</span></span></li>
                    <li><span><kbd>partialCharacter(Ideal)</kbd> -- see <span><a title="Computing the partial character of a cellular binomial ideal" href="_partial__Character.html">partialCharacter</a> -- Computing the partial character of a cellular binomial ideal</span></span></li>
                  </ul>
                </li>
                <li>Symbols                  <ul>
                    <li><span><a title="cellular variables" href="___Cell__Variables.html">CellVariables</a> -- cellular variables</span></li>
                    <li><span><a title="return the cellular variables" href="___Return__Cell__Vars.html">ReturnCellVars</a> -- return the cellular variables</span></li>
                    <li><span><a title="return two partial characters" href="___Return__P__Chars.html">ReturnPChars</a> -- return two partial characters</span></li>
                    <li><span><a title="return two associated primes" href="___Return__Primes.html">ReturnPrimes</a> -- return two associated primes</span></li>
                  </ul>
                </li>
              </ul>
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        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="a package for binomial ideals" href="index.html">Binomials</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">Binomials.m2</span>.</p>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">Binomials.m2:1403:0</span>.</p>
        </div>
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