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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_rings.html" title="">rings</a> > <a href="_monomial_sporderings.html" title="">monomial orderings</a> > <a href="___Monomial__Order.html" title="monomial ordering">MonomialOrder</a></div>
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<div><h1>MonomialOrder -- monomial ordering</h1>
<div class="single"><h2>Description</h2>
<div><tt>MonomialOrder</tt> -- an optional argument used with polynomial rings and monoids to indicate a monomial ordering other than the default (graded reverse lexicographic).<p/>
In Macaulay2, each polynomial ring (and also each monoid) is equipped with a monomial order, which is used for display of polynomials (terms are listed in descending monomial order), and also for Gröbner basis computations.<p/>
In the most general setting, a monomial ordering is given by a list of <em>ordering tests</em>, of various types listed and described below, each of which provides a partial ordering on the monomials. The ordering tests are applied sequentially, starting with the first one, until one monomial is judged greater than the other. At the end, if necessary, the graded reverse lexicographic order is used to compare the monomials. For examples, see below, or see <a href="_monomial_sporderings.html" title="">monomial orderings</a>.<p/>
Permissible elements:<ul><li><a href="___G__Rev__Lex.html" title="graded reverse lexicographical monomial order.">GRevLex</a> => n -- A graded reverse lexicographic block of variables</li>
<li><a href="___Lex.html" title="lexicographical monomial order.">Lex</a> => n</li>
<li><a href="___Weights.html" title="assigning weights to the variables">Weights</a> => {...}</li>
<li><a href="___Monomial__Order.html" title="monomial ordering">Position</a> => Up or Position => Down</li>
<li><a href="___Rev__Lex.html" title="reverse lexicographic ordering">RevLex</a> => n</li>
<li><a href="___Group__Lex.html" title="defines a ring where some variables are inverted">GroupLex</a> => n</li>
<li><a href="___Group__Rev__Lex.html" title="">GroupRevLex</a> => n</li>
<li><a href="___Monomial__Size.html" title="name for an optional argument">MonomialSize</a> => n, n being 8,16,32, or 64. Set the packing size for exponents for further variables</li>
</ul>
<p/>
Some examples of monomial orders. Note that if only one item is in the list, we can dispense with the list.<ul><li><tt>MonomialOrder => {GRevLex=>2, GRevLex=>3}</tt> -- a product order</li>
<li><tt>MonomialOrder => {2, 3}</tt> -- same</li>
<li><tt>MonomialOrder => {Weights=>{1,13,6,2}}</tt> -- a weight order</li>
<li><tt>MonomialOrder => Weights=>{1,13,6,2}</tt> -- same</li>
</ul>
If any monomials will be less than 1 in the ordering, then the option <tt>Global => false</tt> should be used.<table class="examples"><tr><td><pre>i1 : QQ[x,y, Weights => {-1,1}, Global => false]
o1 = QQ[x, y]
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : x<1
o2 = true</pre>
</td></tr>
<tr><td><pre>i3 : y<1
o3 = false</pre>
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<div class="single"><h2>Functions with optional argument named MonomialOrder :</h2>
<ul><li><span>graphIdeal(..., MonomialOrder => ...), see <span><a href="_graph__Ideal_lp__Ring__Map_rp.html" title="the ideal of the graph of the regular map corresponding to a ring map">graphIdeal(RingMap)</a> -- the ideal of the graph of the regular map corresponding to a ring map</span></span></li>
<li><span>graphRing(..., MonomialOrder => ...), see <span><a href="_graph__Ring_lp__Ring__Map_rp.html" title="the coordinate ring of the graph of the regular map corresponding to a ring map">graphRing(RingMap)</a> -- the coordinate ring of the graph of the regular map corresponding to a ring map</span></span></li>
<li><span>monoid(..., MonomialOrder => ...), see <span><a href="_monoid.html" title="make or retrieve a monoid">monoid</a> -- make or retrieve a monoid</span></span></li>
<li><span>newRing(..., MonomialOrder => ...), see <span><a href="_new__Ring.html" title="make a copy of a ring, with some features changed">newRing</a> -- make a copy of a ring, with some features changed</span></span></li>
<li><span>pushForward(..., MonomialOrder => ...), see <span><a href="_push__Forward_lp__Ring__Map_cm__Module_rp.html" title="">pushForward(RingMap,Module)</a></span></span></li>
<li><span><a href="_rsort_lp..._cm_sp__Monomial__Order_sp_eq_gt_sp..._rp.html" title="specify Ascending or Descending monomial order">rsort(..., MonomialOrder => ...)</a> -- specify Ascending or Descending monomial order</span></li>
<li><span><a href="_sort_lp..._cm_sp__Monomial__Order_sp_eq_gt_sp..._rp.html" title="specify Ascending or Descending monomial order">sort(..., MonomialOrder => ...)</a> -- specify Ascending or Descending monomial order</span></li>
<li><span><a href="_sort__Columns_lp..._cm_sp__Monomial__Order_sp_eq_gt_sp..._rp.html" title="specify Ascending or Descending monomial order">sortColumns(..., MonomialOrder => ...)</a> -- specify Ascending or Descending monomial order</span></li>
<li><span>symmetricAlgebra(..., MonomialOrder => ...), see <span><a href="_symmetric__Algebra.html" title="the symmetric algebra of a module">symmetricAlgebra</a> -- the symmetric algebra of a module</span></span></li>
<li><span>tensor(..., MonomialOrder => ...), see <span><a href="_tensor_lp__Ring_cm__Ring_rp.html" title="tensor product">tensor(Ring,Ring)</a> -- tensor product</span></span></li>
</ul>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_monomial_sporderings.html" title="">monomial orderings</a></span></li>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Monomial__Order.html" title="monomial ordering">MonomialOrder</a> is <span>a <a href="___Symbol.html">symbol</a></span>.</p>
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