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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_sporders_spfor_spfree_spmodules.html" title="">monomial orders for free modules</a></div>
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<div><h1>monomial orders for free modules</h1>
<div>In Macaulay2, each free module <i>F = R<sup>s</sup></i> over a ring <i>R</i> has a basis of unit column vectors <i>F<sub>0</sub>, F<sub>1</sub>, ..., F<sub>(</sub>s-1)</i>. The monomials of <i>F</i> are the elements <i>m F<sub>i</sub></i>, where <i>m</i> is a monomial of the ring <i>R</i>. In Macaulay2, orders on the monomials of <i>F</i> are used for computing Gröbner bases and syzygies, and also to determine the initial, or lead term of elements of <i>F</i>.<p/>
The ring <i>R</i> comes equipped with a total order on the monomials of <i>R</i>. A total order on the monomials of <i>F</i> is called <b>compatible</b> (with the order on <i>R</i>), if <i>m F<sub>i</sub> > n F<sub>i</sub></i> (in <i>F</i>) whenever <i>m > n</i> (in <i>R</i>). There are many types of compatible orders, but several stand out: term over position up (the default in Macaulay2), term over position down, position up over term, position down over term, and Schreyer orders.<p/>
term over position up: <i>m F<sub>i</sub> > n F<sub>j</sub></i> iff <i>m>n</i> or <i>m==n</i> and <i>i>j</i><p/>
term over position down: <i>m F<sub>i</sub> > n F<sub>j</sub></i> iff <i>m>n</i> or <i>m==n</i> and <i>i<j</i><p/>
position up over term: <i>m F<sub>i</sub> > n F<sub>j</sub></i> iff <i>i>j</i> or <i>i==j</i> and <i>m>n</i><p/>
position down over term: <i>m F<sub>i</sub> > n F<sub>j</sub></i> iff <i>i<j</i> or <i>i==j</i> and <i>m>n</i><p/>
Induced monomial orders are another class of important orders on <tt>F</tt>, see <a href="___Schreyer_sporders.html" title="induced monomial order on a free module">Schreyer orders</a> for their definition and use in Macaulay2.<p/>
In Macaulay2, free modules come equipped with a compatible order. The default order is: term over position up. This is called Position=>Up. In the following example, the lead term is <i>a F<sub>1</sub></i>, since <i>a > b</i>.<table class="examples"><tr><td><pre>i1 : R = ZZ[a..d];</pre>
</td></tr>
<tr><td><pre>i2 : F = R^3
3
o2 = R
o2 : R-module, free</pre>
</td></tr>
<tr><td><pre>i3 : f = b*F_0 + a*F_1
o3 = | b |
| a |
| 0 |
3
o3 : R</pre>
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<tr><td><pre>i4 : leadTerm f
o4 = | 0 |
| a |
| 0 |
3
o4 : R</pre>
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</table>
This is the same as giving the monomial order as:<table class="examples"><tr><td><pre>i5 : R = ZZ[a..d, MonomialOrder => {GRevLex => 4, Position => Up}];</pre>
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<tr><td><pre>i6 : F = R^3
3
o6 = R
o6 : R-module, free</pre>
</td></tr>
<tr><td><pre>i7 : leadTerm(a*F_0 + a*F_1)
o7 = | 0 |
| a |
| 0 |
3
o7 : R</pre>
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</table>
Giving Position=>Down instead switches the test above to i < j. In this case the monomial order on F is: m*F_i > n*F_j if m>n or m==n and i<j.<table class="examples"><tr><td><pre>i8 : R = ZZ[a..d, MonomialOrder => {GRevLex => 4, Position => Down}];</pre>
</td></tr>
<tr><td><pre>i9 : F = R^3
3
o9 = R
o9 : R-module, free</pre>
</td></tr>
<tr><td><pre>i10 : leadTerm(a*F_0 + a*F_1)
o10 = | a |
| 0 |
| 0 |
3
o10 : R</pre>
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</table>
If one gives Position=>Up or Position=>Down earlier, then the position will be taken into account earlier. For example<table class="examples"><tr><td><pre>i11 : R = ZZ[a..d, MonomialOrder => {GRevLex => 2, Position => Down, GRevLex => 2}];</pre>
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<tr><td><pre>i12 : F = R^3
3
o12 = R
o12 : R-module, free</pre>
</td></tr>
<tr><td><pre>i13 : leadTerm(a*F_0 + a*F_1)
o13 = | a |
| 0 |
| 0 |
3
o13 : R</pre>
</td></tr>
<tr><td><pre>i14 : leadTerm(b*F_0 + c^4*F_1)
o14 = | b |
| 0 |
| 0 |
3
o14 : R</pre>
</td></tr>
<tr><td><pre>i15 : leadTerm(c*F_0 + d^2*F_1)
o15 = | c |
| 0 |
| 0 |
3
o15 : R</pre>
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</table>
If one wants Position over Term (POT), place the Position element first<table class="examples"><tr><td><pre>i16 : R = ZZ[a..d, MonomialOrder => {Position => Down}];</pre>
</td></tr>
<tr><td><pre>i17 : F = R^3
3
o17 = R
o17 : R-module, free</pre>
</td></tr>
<tr><td><pre>i18 : leadTerm(a*F_0 + a*F_1)
o18 = | a |
| 0 |
| 0 |
3
o18 : R</pre>
</td></tr>
<tr><td><pre>i19 : leadTerm(b*F_0 + c^4*F_1)
o19 = | b |
| 0 |
| 0 |
3
o19 : R</pre>
</td></tr>
<tr><td><pre>i20 : leadTerm(c*F_0 + d^2*F_1)
o20 = | c |
| 0 |
| 0 |
3
o20 : R</pre>
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<div><h3>Menu</h3>
<ul><li><span><a href="___Schreyer_sporders.html" title="induced monomial order on a free module">Schreyer orders</a> -- induced monomial order on a free module</span></li>
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