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<head><title>* -- a binary operator, usually used for multiplication</title>
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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="___The_sp__Macaulay2_splanguage.html" title="">The Macaulay2 language</a> > <a href="_operators.html" title="">operators</a> > <a href="__st.html" title="a binary operator, usually used for multiplication">*</a></div>
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<div><h1>* -- a binary operator, usually used for multiplication</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>x * y</tt></div>
</dd></dl>
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</li>
</ul>
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<div class="single"><h2>Description</h2>
<div>The return type depends on the types of x and y. If they have the same type, then usually the return type is the common type of x and y.<p/>
Multiplication involving ring elements (including integers, rational numbers, real and complex numbers), ideals, vectors, matrices, modules is generally the usual multiplication, or composition of functions.<p/>
The intersection of sets is given by multiplication. See <a href="___Set_sp_st_sp__Set.html" title="intersection of sets">Set * Set</a>.<table class="examples"><tr><td><pre>i1 : set{hi,you,there} * set{hi,us,here,you}
o1 = set {hi, you}
o1 : Set</pre>
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<p/>
Multiplication involving a list attempts to multiply each element of the list.<table class="examples"><tr><td><pre>i2 : R = QQ[a..d];</pre>
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<tr><td><pre>i3 : a * {b,c,d}
o3 = {a*b, a*c, a*d}
o3 : List</pre>
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<p/>
Multiplication of matrices (<a href="___Matrix_sp_st_sp__Matrix.html" title="matrix multiplication">Matrix * Matrix</a>) or ring maps is the same as composition.<table class="examples"><tr><td><pre>i4 : f = map(R,R,{b,c,a,d})
o4 = map(R,R,{b, c, a, d})
o4 : RingMap R <--- R</pre>
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<tr><td><pre>i5 : g = map(R,R,{(a+b)^2,b^2,c^2,d^2})
2 2 2 2 2
o5 = map(R,R,{a + 2a*b + b , b , c , d })
o5 : RingMap R <--- R</pre>
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<tr><td><pre>i6 : f*g
2 2 2 2 2
o6 = map(R,R,{b + 2b*c + c , c , a , d })
o6 : RingMap R <--- R</pre>
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<tr><td><pre>i7 : (f*g)(a) == f(g(a))
o7 = true</pre>
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<p/>
Submodules of modules may be produced using multiplication and addition.<table class="examples"><tr><td><pre>i8 : M = R^2; I = ideal(a+b,c);
o9 : Ideal of R</pre>
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<tr><td><pre>i10 : N = I*M + a*R^2
o10 = image | a+b 0 c 0 a 0 |
| 0 a+b 0 c 0 a |
2
o10 : R-module, submodule of R</pre>
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<tr><td><pre>i11 : isHomogeneous N
o11 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_times.html" title="multiplication">times</a> -- multiplication</span></li>
<li><span><a href="_product.html" title="">product</a></span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>*</tt> :</h2>
<ul><li>AffineVariety * AffineVariety</li>
<li>CC * CC</li>
<li>CC * QQ</li>
<li>CC * RR</li>
<li>CC * ZZ</li>
<li>ChainComplexMap * ChainComplexMap</li>
<li>Constant * Constant</li>
<li>Constant * InexactNumber</li>
<li>Constant * Number</li>
<li>GradedModuleMap * GradedModuleMap</li>
<li>Ideal * CoherentSheaf</li>
<li>Ideal * Module</li>
<li>Ideal * Ring</li>
<li>Ideal * Vector</li>
<li>InexactNumber * Constant</li>
<li>Matrix * Number</li>
<li>Matrix * RingElement</li>
<li>Matrix * Vector</li>
<li>Matrix * ZZ</li>
<li>MonomialIdeal * Module</li>
<li>MonomialIdeal * MonomialIdeal</li>
<li>MonomialIdeal * Ring</li>
<li>MutableMatrix * MutableMatrix</li>
<li>Number * Constant</li>
<li>Number * Matrix</li>
<li>QQ * CC</li>
<li>QQ * QQ</li>
<li>QQ * RR</li>
<li>QQ * ZZ</li>
<li>Ring * Ideal</li>
<li>Ring * MonomialIdeal</li>
<li>Ring * RingElement</li>
<li>Ring * Vector</li>
<li>RingElement * ChainComplexMap</li>
<li>RingElement * GradedModuleMap</li>
<li>RingElement * Ideal</li>
<li>RingElement * Matrix</li>
<li>RingElement * Module</li>
<li>RingElement * MonomialIdeal</li>
<li>RingElement * MutableMatrix</li>
<li>RingElement * RingElement</li>
<li>RingElement * Vector</li>
<li>RingMap * RingMap</li>
<li>RR * CC</li>
<li>RR * QQ</li>
<li>RR * RR</li>
<li>RR * ZZ</li>
<li>Thing * List</li>
<li>ZZ * CC</li>
<li>ZZ * ProjectiveHilbertPolynomial</li>
<li>ZZ * QQ</li>
<li>ZZ * RR</li>
<li>ZZ * ZZ</li>
<li><span>QQ * BettiTally, see <span><a href="___Betti__Tally.html" title="the class of all Betti tallies">BettiTally</a> -- the class of all Betti tallies</span></span></li>
<li><span>ZZ * BettiTally, see <span><a href="___Betti__Tally.html" title="the class of all Betti tallies">BettiTally</a> -- the class of all Betti tallies</span></span></li>
<li><span>Constant * RingElement, see <span><a href="___Constant.html" title="">Constant</a></span></span></li>
<li><span>RingElement * Constant, see <span><a href="___Constant.html" title="">Constant</a></span></span></li>
<li><span>Expression * Expression, see <span><a href="___Expression.html" title="the class of all expressions">Expression</a> -- the class of all expressions</span></span></li>
<li><span><a href="___Ideal_sp_st_sp__Ideal.html" title="product of ideals">Ideal * Ideal</a> -- product of ideals</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><tt>InexactNumber * RingElement</tt> (missing documentation<!-- tag: (*,InexactNumber,RingElement) -->)</span></li>
<li><span><a href="___Matrix_sp_st_sp__Matrix.html" title="matrix multiplication">Matrix * Matrix</a> -- matrix multiplication</span></li>
<li><span><tt>RingElement * InexactNumber</tt> (missing documentation<!-- tag: (*,RingElement,InexactNumber) -->)</span></li>
<li><span><a href="___Set_sp_st_sp__Set.html" title="intersection of sets">Set * Set</a> -- intersection of sets</span></li>
</ul>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="__st.html" title="a binary operator, usually used for multiplication">*</a> is <span>a <a href="___Keyword.html">keyword</a></span>.</p>
<div><div><p>This operator may be used as a binary operator in an expression like <tt>x*y</tt>. The user may install <a href="_binary_spmethods.html" title="">binary methods</a> for handling such expressions with code such as</p>
<pre> X * Y := (x,y) -> ...</pre>
<p>where <tt>X</tt> is the class of <tt>x</tt> and <tt>Y</tt> is the class of <tt>y</tt>.</p>
</div>
<div><p>This operator may be used as a prefix unary operator in an expression like <tt>*y</tt>. The user may <a href="_installing_spmethods.html">install a method</a> for handling such expressions with code such as</p>
<pre> * Y := (y) -> ...</pre>
<p>where <tt>Y</tt> is the class of <tt>y</tt>.</p>
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