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<head><title>definition of product (block) orders</title>
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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="_definition_spof_spproduct_sp_lpblock_rp_sporders.html" title="">definition of product (block) orders</a></div>
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<div><h1>definition of product (block) orders</h1>
<div><tt>MonomialOrder => {n_1, ..., n_l}</tt> divides the variables of the ring into <tt>l</tt> blocks, the first block consisting of the first <tt>n_1</tt> variables, the second block consisting of the subsequent <tt>n_2</tt> variables, and so on. For each block of variables, we can compute the total degree of a monomial with respect to the variables in that block. This gives a length <tt>l</tt> vector of total degrees for each monomial. We say x^A > x^B if the total degree vector of x^A is lexicographically greater than the total degree vector of x^B, or if the two total degree vectors are equal and if in the first block of variables where A and B differ, A > B in GRevLex order.<table class="examples"><tr><td><pre>i1 : R = QQ[a..l, MonomialOrder => {3,3,3,3}];</pre>
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<tr><td><pre>i2 : a*e^3 + a^2*c*i + a*b^2*i + b^2*e*i
2 2 2 3
o2 = a*b i + a c*i + b e*i + a*e
o2 : R</pre>
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We may replace <tt>MonomialOrder => {3,3,3,3}</tt> with the shorter <tt>MonomialOrder => {4:3}</tt><p/>
The default <tt>GRevLex</tt> order on any block may be changed to other orders, as follows.<table class="examples"><tr><td><pre>i3 : R = QQ[a..i, MonomialOrder => {Lex =>3,3:1,3}];</pre>
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<tr><td><pre>i4 : a*e^3 + a^2*c*i + a*b^2*i + b^2*e*i + d^2*f*h + d*e^2*h
2 2 3 2 2 2
o4 = a c*i + a*b i + a*e + b e*i + d f*h + d*e h
o4 : R</pre>
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Note: <tt>Weights</tt> and <tt>Eliminate</tt> do not create blocks, they only assign weights to the variables.</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Weights.html" title="assigning weights to the variables">Weights</a> -- assigning weights to the variables</span></li>
<li><span><a href="___Eliminate.html" title="elimination order">Eliminate</a> -- elimination order</span></li>
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