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<head><title>MonomialIdeal -- the class of all monomial ideals handled by the engine</title>
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<div><h1>MonomialIdeal -- the class of all monomial ideals handled by the engine</h1>
<div class="single"><h2>Description</h2>
<div>Monomial ideals are kinds of ideals, but many algorithms are much faster. Generally, any routines available for ideals are also available for monomial ideals.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
</td></tr>
<tr><td><pre>i2 : I = monomialIdeal(a*b*c,b*c*d,a^2*d,b^3*c)
3 2
o2 = monomialIdeal (a*b*c, b c, a d, b*c*d)
o2 : MonomialIdeal of R</pre>
</td></tr>
<tr><td><pre>i3 : I^2
2 2 2 4 2 6 2 3 2 3 2 2 4 2
o3 = monomialIdeal (a b c , a*b c , b c , a b*c*d, a b c*d, a*b c d, b c d,
------------------------------------------------------------------------
4 2 2 2 2 2 2
a d , a b*c*d , b c d )
o3 : MonomialIdeal of R</pre>
</td></tr>
<tr><td><pre>i4 : I + monomialIdeal(b*c)
2
o4 = monomialIdeal (b*c, a d)
o4 : MonomialIdeal of R</pre>
</td></tr>
<tr><td><pre>i5 : I : monomialIdeal(b*c)
2
o5 = monomialIdeal (a, b , d)
o5 : MonomialIdeal of R</pre>
</td></tr>
<tr><td><pre>i6 : radical I
o6 = monomialIdeal (b*c, a*d)
o6 : MonomialIdeal of R</pre>
</td></tr>
<tr><td><pre>i7 : associatedPrimes I
o7 = {monomialIdeal (a, b), monomialIdeal (a, c), monomialIdeal (b, d),
------------------------------------------------------------------------
monomialIdeal (c, d), monomialIdeal (a, b, d)}
o7 : List</pre>
</td></tr>
<tr><td><pre>i8 : primaryDecomposition I
2 2
o8 = {monomialIdeal (a , b), monomialIdeal (a , c), monomialIdeal (b, d),
------------------------------------------------------------------------
3
monomialIdeal (c, d), monomialIdeal (a, b , d)}
o8 : List</pre>
</td></tr>
</table>
<h3>Specialized functions only available for monomial ideals</h3>
<ul><li><span><a href="_borel_lp__Matrix_rp.html" title="make a Borel fixed submodule">borel(MonomialIdeal)</a> -- make a Borel fixed submodule</span></li>
<li><span><a href="_is__Borel.html" title="whether an ideal is fixed by upper triangular changes of coordinates">isBorel(MonomialIdeal)</a> -- whether an ideal is fixed by upper triangular changes of coordinates</span></li>
<li><span><a href="___Monomial__Ideal_sp-_sp__Monomial__Ideal.html" title="monomial ideal difference">MonomialIdeal - MonomialIdeal</a> -- monomial ideal difference</span></li>
<li><span><a href="_dual_lp__Monomial__Ideal_rp.html" title="the Alexander dual of a monomial ideal">dual(MonomialIdeal)</a> -- the Alexander dual of a monomial ideal</span></li>
<li><span><a href="_independent__Sets.html" title="some size-maximal independent subsets of variables modulo an ideal">independentSets</a> -- some size-maximal independent subsets of variables modulo an ideal</span></li>
<li><span><a href="../../PrimaryDecomposition/html/_irreducible__Decomposition_lp__Monomial__Ideal_rp.html" title="express a monomial ideal as an intersection of irreducible monomial ideals">irreducibleDecomposition</a> -- express a monomial ideal as an intersection of irreducible monomial ideals</span></li>
<li><span><a href="_standard__Pairs.html" title="find the standard pairs of a monomial ideal">standardPairs</a> -- find the standard pairs of a monomial ideal</span></li>
</ul>
<table class="examples"><tr><td><pre>i9 : borel I
3 2 2 3 2 2 2 2 2
o9 = monomialIdeal (a , a b, a*b , b , a c, a*b*c, b c, a*c , b*c , a d,
------------------------------------------------------------------------
2
a*b*d, b d, a*c*d, b*c*d)
o9 : MonomialIdeal of R</pre>
</td></tr>
<tr><td><pre>i10 : isBorel I
o10 = false</pre>
</td></tr>
<tr><td><pre>i11 : I - monomialIdeal(b^3*c,b^4)
2
o11 = monomialIdeal (a*b*c, a d, b*c*d)
o11 : MonomialIdeal of R</pre>
</td></tr>
<tr><td><pre>i12 : standardPairs I
o12 = {{1, {c, d}}, {a, {c, d}}, {1, {b, d}}, {a, {b, d}}, {1, {c, a}}, {1,
-----------------------------------------------------------------------
2
{b, a}}, {b, {c}}, {b , {c}}}
o12 : List</pre>
</td></tr>
<tr><td><pre>i13 : independentSets I
o13 = {a*b, a*c, b*d, c*d}
o13 : List</pre>
</td></tr>
<tr><td><pre>i14 : dual I
3 2 3
o14 = monomialIdeal (a*b , a*c, a b*d, b d, c*d)
o14 : MonomialIdeal of R</pre>
</td></tr>
</table>
The ring of a monomial ideal must be a commutative polynomial ring. This ring must not be a skew commuting ring, and/or a quotient ring.</div>
</div>
<div class="waystouse"><h2>Functions and methods returning a monomial ideal :</h2>
<ul><li><span>MonomialIdeal * MonomialIdeal, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>RingElement * MonomialIdeal, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>MonomialIdeal + MonomialIdeal, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>borel(MonomialIdeal), see <span><a href="_borel_lp__Matrix_rp.html" title="make a Borel fixed submodule">borel(Matrix)</a> -- make a Borel fixed submodule</span></span></li>
<li><span>MonomialIdeal ^ ZZ, see <span><a href="___Ideal_sp^_sp__Z__Z.html" title="power">Ideal ^ ZZ</a> -- power</span></span></li>
<li><span><a href="_monomial__Ideal.html" title="make a monomial ideal">monomialIdeal</a> -- make a monomial ideal</span></li>
<li><span><a href="___Monomial__Ideal_sp-_sp__Monomial__Ideal.html" title="monomial ideal difference">MonomialIdeal - MonomialIdeal</a> -- monomial ideal difference</span></li>
<li><span><a href="_monomial__Ideal_lp__Ideal_rp.html" title="monomial ideal of lead monomials of a Gröbner basis">monomialIdeal(Ideal)</a> -- monomial ideal of lead monomials of a Gröbner basis</span></li>
<li><span>monomialIdeal(Module), see <span><a href="_monomial__Ideal_lp__Ideal_rp.html" title="monomial ideal of lead monomials of a Gröbner basis">monomialIdeal(Ideal)</a> -- monomial ideal of lead monomials of a Gröbner basis</span></span></li>
<li><span>monomialIdeal(List), see <span><a href="_monomial__Ideal_lp__Matrix_rp.html" title="monomial ideal of lead monomials">monomialIdeal(Matrix)</a> -- monomial ideal of lead monomials</span></span></li>
<li><span><a href="_monomial__Ideal_lp__Matrix_rp.html" title="monomial ideal of lead monomials">monomialIdeal(Matrix)</a> -- monomial ideal of lead monomials</span></li>
<li><span>monomialIdeal(RingElement), see <span><a href="_monomial__Ideal_lp__Matrix_rp.html" title="monomial ideal of lead monomials">monomialIdeal(Matrix)</a> -- monomial ideal of lead monomials</span></span></li>
<li><span><a href="../../Classic/html/_monomial__Ideal_lp__String_rp.html" title="make a monomial ideal using classic Macaulay syntax">monomialIdeal(String)</a> -- make a monomial ideal using classic Macaulay syntax</span></li>
<li><span>MonomialIdeal : MonomialIdeal, see <span><a href="_quotient_lp__Ideal_cm__Ideal_rp.html" title="ideal or submodule quotient">quotient(Ideal,Ideal)</a> -- ideal or submodule quotient</span></span></li>
<li><span>quotient(MonomialIdeal,MonomialIdeal), see <span><a href="_quotient_lp__Ideal_cm__Ideal_rp.html" title="ideal or submodule quotient">quotient(Ideal,Ideal)</a> -- ideal or submodule quotient</span></span></li>
<li><span>radical(MonomialIdeal), see <span><a href="_radical.html" title="the radical of an ideal">radical</a> -- the radical of an ideal</span></span></li>
<li><span>saturate(MonomialIdeal,MonomialIdeal), see <span><a href="_saturate.html" title="saturation of ideal or submodule">saturate</a> -- saturation of ideal or submodule</span></span></li>
</ul>
<h2>Methods that use a monomial ideal :</h2>
<ul><li><span>ZZ % MonomialIdeal, see <span><a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a> -- a binary operator, usually used for remainder and reduction</span></span></li>
<li><span>MonomialIdeal * Module, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>MonomialIdeal * Ring, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>Ring * MonomialIdeal, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>ZZ // MonomialIdeal, see <span><a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> -- a binary operator, usually used for quotient</span></span></li>
<li><span>Ideal == MonomialIdeal, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>MonomialIdeal == Ideal, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>MonomialIdeal == MonomialIdeal, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>MonomialIdeal == Ring, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>MonomialIdeal == ZZ, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>Ring == MonomialIdeal, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>ZZ == MonomialIdeal, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>associatedPrimes(MonomialIdeal), see <span><a href="../../PrimaryDecomposition/html/_associated__Primes_lp__Ideal_rp.html" title="find the associated primes of an ideal">associatedPrimes(Ideal)</a> -- find the associated primes of an ideal</span></span></li>
<li><span>betti(MonomialIdeal), see <span><a href="_betti_lp__Ideal_rp.html" title="gives the degrees of generators.">betti(Ideal)</a> -- gives the degrees of generators.</span></span></li>
<li><span><a href="_codim_lp__Monomial__Ideal_rp.html" title="compute the codimension">codim(MonomialIdeal)</a> -- compute the codimension</span></li>
<li><span>dim(MonomialIdeal), see <span><a href="_dim_lp__Ideal_rp.html" title="compute the Krull dimension">dim(Ideal)</a> -- compute the Krull dimension</span></span></li>
<li><span><a href="_dual_lp__Monomial__Ideal_rp.html" title="the Alexander dual of a monomial ideal">dual(MonomialIdeal)</a> -- the Alexander dual of a monomial ideal</span></li>
<li><span><a href="_dual_lp__Monomial__Ideal_cm__List_rp.html" title="the Alexander dual">dual(MonomialIdeal,List)</a> -- the Alexander dual</span></li>
<li><span><a href="_dual_lp__Monomial__Ideal_cm__Ring__Element_rp.html" title="the Alexander dual">dual(MonomialIdeal,RingElement)</a> -- the Alexander dual</span></li>
<li><span>MonomialIdeal _ ZZ, see <span><a href="_generators_spof_spideals_spand_spmodules.html" title="">generators of ideals and modules</a></span></span></li>
<li><span>generators(MonomialIdeal), see <span><a href="_generators_lp__Ideal_rp.html" title="the generator matrix of an ideal">generators(Ideal)</a> -- the generator matrix of an ideal</span></span></li>
<li><span>Ideal * MonomialIdeal, see <span><a href="___Ideal_sp_st_sp__Ideal.html" title="product of ideals">Ideal * Ideal</a> -- product of ideals</span></span></li>
<li><span>MonomialIdeal * Ideal, see <span><a href="___Ideal_sp_st_sp__Ideal.html" title="product of ideals">Ideal * Ideal</a> -- product of ideals</span></span></li>
<li><span>Ideal + MonomialIdeal, see <span><a href="___Ideal_sp_pl_sp__Ideal.html" title="sum of ideals">Ideal + Ideal</a> -- sum of ideals</span></span></li>
<li><span>MonomialIdeal + Ideal, see <span><a href="___Ideal_sp_pl_sp__Ideal.html" title="sum of ideals">Ideal + Ideal</a> -- sum of ideals</span></span></li>
<li><span><a href="_ideal_lp__Monomial__Ideal_rp.html" title="converts a monomial ideal to an ideal">ideal(MonomialIdeal)</a> -- converts a monomial ideal to an ideal</span></li>
<li><span>independentSets(MonomialIdeal), see <span><a href="_independent__Sets.html" title="some size-maximal independent subsets of variables modulo an ideal">independentSets</a> -- some size-maximal independent subsets of variables modulo an ideal</span></span></li>
<li><span><a href="../../PrimaryDecomposition/html/_irreducible__Decomposition_lp__Monomial__Ideal_rp.html" title="express a monomial ideal as an intersection of irreducible monomial ideals">irreducibleDecomposition(MonomialIdeal)</a> -- express a monomial ideal as an intersection of irreducible monomial ideals</span></li>
<li><span>isBorel(MonomialIdeal), see <span><a href="_is__Borel.html" title="whether an ideal is fixed by upper triangular changes of coordinates">isBorel</a> -- whether an ideal is fixed by upper triangular changes of coordinates</span></span></li>
<li><span>isIdeal(MonomialIdeal), see <span><a href="_is__Ideal.html" title="whether something is an ideal">isIdeal</a> -- whether something is an ideal</span></span></li>
<li><span>isMonomialIdeal(MonomialIdeal), see <span><a href="_is__Monomial__Ideal.html" title="whether something is a monomial ideal">isMonomialIdeal</a> -- whether something is a monomial ideal</span></span></li>
<li><span>isSquareFree(MonomialIdeal), see <span><a href="_is__Square__Free.html" title="whether something is square free monomial ideal">isSquareFree</a> -- whether something is square free monomial ideal</span></span></li>
<li><span>jacobian(MonomialIdeal), see <span><a href="_jacobian_lp__Ideal_rp.html" title="the Jacobian matrix of the generators of an ideal">jacobian(Ideal)</a> -- the Jacobian matrix of the generators of an ideal</span></span></li>
<li><span><a href="_lcm_lp__Monomial__Ideal_rp.html" title="least common multiple of all minimal generators">lcm(MonomialIdeal)</a> -- least common multiple of all minimal generators</span></li>
<li><span>Matrix // MonomialIdeal, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span>RingElement // MonomialIdeal, see <span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></span></li>
<li><span>Matrix % MonomialIdeal, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>RingElement % MonomialIdeal, see <span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">methods for normal forms and remainder</a> -- calculate the normal form of ring elements and matrices</span></span></li>
<li><span>decompose(MonomialIdeal), see <span><a href="_minimal__Primes.html" title="minimal associated primes of an ideal">minimalPrimes</a> -- minimal associated primes of an ideal</span></span></li>
<li><span>minimalPrimes(MonomialIdeal), see <span><a href="_minimal__Primes.html" title="minimal associated primes of an ideal">minimalPrimes</a> -- minimal associated primes of an ideal</span></span></li>
<li><span>module(MonomialIdeal), see <span><a href="_module_lp__Ideal_rp.html" title="turn an ideal into a module">module(Ideal)</a> -- turn an ideal into a module</span></span></li>
<li><span>numgens(MonomialIdeal), see <span><a href="_numgens_lp__Ring_rp.html" title="number of generators of a polynomial ring">numgens(Ring)</a> -- number of generators of a polynomial ring</span></span></li>
<li><span>primaryDecomposition(MonomialIdeal), see <span><a href="../../PrimaryDecomposition/html/_primary__Decomposition.html" title="irredundant primary decomposition of an ideal">primaryDecomposition</a> -- irredundant primary decomposition of an ideal</span></span></li>
<li><span>resolution(MonomialIdeal), see <span><a href="_resolution_lp__Ideal_rp.html" title="compute a projective resolution of (the quotient ring corresponding to) an ideal">resolution(Ideal)</a> -- compute a projective resolution of (the quotient ring corresponding to) an ideal</span></span></li>
<li><span>ring(MonomialIdeal), see <span><a href="_ring.html" title="get the associated ring of an object">ring</a> -- get the associated ring of an object</span></span></li>
<li><span>Ring / MonomialIdeal, see <span><a href="___Ring_sp_sl_sp__Ideal.html" title="make a quotient ring">Ring / Ideal</a> -- make a quotient ring</span></span></li>
<li><span>saturate(MonomialIdeal,RingElement), see <span><a href="_saturate.html" title="saturation of ideal or submodule">saturate</a> -- saturation of ideal or submodule</span></span></li>
<li><span>standardPairs(MonomialIdeal), see <span><a href="_standard__Pairs.html" title="find the standard pairs of a monomial ideal">standardPairs</a> -- find the standard pairs of a monomial ideal</span></span></li>
<li><span>standardPairs(MonomialIdeal,List), see <span><a href="_standard__Pairs.html" title="find the standard pairs of a monomial ideal">standardPairs</a> -- find the standard pairs of a monomial ideal</span></span></li>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Monomial__Ideal.html" title="the class of all monomial ideals handled by the engine">MonomialIdeal</a> is <span>a <a href="___Type.html">type</a></span>, with ancestor classes <a href="___Ideal.html" title="the class of all ideals">Ideal</a> < <a href="___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="___Thing.html" title="the class of all things">Thing</a>.</p>
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