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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="__pl_pl.html" title="a binary operator, usually used for direct sum">++</a></div>
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<div><h1>++ -- a binary operator, usually used for direct sum</h1>
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<div class="waystouse"><h2>Ways to use <tt>++</tt> :</h2>
<ul><li><span>BettiTally ++ 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><a href="___Chain__Complex_sp_pl_pl_sp__Chain__Complex.html" title="direct sum">ChainComplex ++ ChainComplex</a> -- direct sum</span></li>
<li><span>ChainComplexMap ++ ChainComplexMap, see <span><a href="___Chain__Complex_sp_pl_pl_sp__Chain__Complex.html" title="direct sum">ChainComplex ++ ChainComplex</a> -- direct sum</span></span></li>
<li><span>GradedModule ++ GradedModule, see <span><a href="___Chain__Complex_sp_pl_pl_sp__Chain__Complex.html" title="direct sum">ChainComplex ++ ChainComplex</a> -- direct sum</span></span></li>
<li><span><a href="___Coherent__Sheaf_sp_pl_pl_sp__Coherent__Sheaf.html" title="direct sum of coherent sheaves">CoherentSheaf ++ CoherentSheaf</a> -- direct sum of coherent sheaves</span></li>
<li><span>Option ++ Option, see <span><a href="_direct__Sum.html" title="direct sum of modules or maps">directSum</a> -- direct sum of modules or maps</span></span></li>
<li><span>GradedModuleMap ++ GradedModuleMap, see <span><a href="___Matrix_sp_pl_pl_sp__Matrix.html" title="direct sum of maps">Matrix ++ Matrix</a> -- direct sum of maps</span></span></li>
<li><span><a href="___Matrix_sp_pl_pl_sp__Matrix.html" title="direct sum of maps">Matrix ++ Matrix</a> -- direct sum of maps</span></li>
<li><span>Matrix ++ RingElement, see <span><a href="___Matrix_sp_pl_pl_sp__Matrix.html" title="direct sum of maps">Matrix ++ Matrix</a> -- direct sum of maps</span></span></li>
<li><span>RingElement ++ Matrix, see <span><a href="___Matrix_sp_pl_pl_sp__Matrix.html" title="direct sum of maps">Matrix ++ Matrix</a> -- direct sum of maps</span></span></li>
<li><span>RingElement ++ RingElement, see <span><a href="___Matrix_sp_pl_pl_sp__Matrix.html" title="direct sum of maps">Matrix ++ Matrix</a> -- direct sum of maps</span></span></li>
<li><span>GradedModule ++ Module, see <span><a href="___Module_sp_pl_pl_sp__Module.html" title="direct sum of modules">Module ++ Module</a> -- direct sum of modules</span></span></li>
<li><span>Module ++ GradedModule, see <span><a href="___Module_sp_pl_pl_sp__Module.html" title="direct sum of modules">Module ++ Module</a> -- direct sum of modules</span></span></li>
<li><span><a href="___Module_sp_pl_pl_sp__Module.html" title="direct sum of modules">Module ++ Module</a> -- direct sum of modules</span></li>
<li><span>OptionTable ++ List, see <span><a href="___Option__Table_sp_pl_pl_sp__Option__Table.html" title="">OptionTable ++ OptionTable</a></span></span></li>
<li><span><a href="___Option__Table_sp_pl_pl_sp__Option__Table.html" title="">OptionTable ++ OptionTable</a></span></li>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="__pl_pl.html" title="a binary operator, usually used for direct sum">++</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>
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