https://dictionary.iucr.org/index.php?action=history&feed=atom&title=Binary_operation
Binary operation - Revision history
2026-06-04T02:15:14Z
Revision history for this page on the wiki
MediaWiki 1.30.0
https://dictionary.iucr.org/index.php?title=Binary_operation&diff=4664&oldid=prev
BrianMcMahon: Tidied translations.
2017-11-28T17:03:19Z
<p>Tidied translations.</p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 17:03, 28 November 2017</td>
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<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><font color="blue">Loi de composition</font> (''Fr''). <font color="red">Zweistellige Verknüpfung</font> (''Ge''). <font color="black">Operazione binaria</font> (''It''). <font color="<del class="diffchange diffchange-inline">brown</del>"><del class="diffchange diffchange-inline">Бинарная операция</del></font> (''<del class="diffchange diffchange-inline">Ru</del>''). <font color="<del class="diffchange diffchange-inline">purple</del>"><del class="diffchange diffchange-inline">二項演算</del></font> (''<del class="diffchange diffchange-inline">Ja</del>''). <font color="green">Operación binaria</font> (''Sp'').</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><font color="blue">Loi de composition</font> (''Fr''). <font color="red">Zweistellige Verknüpfung</font> (''Ge''). <font color="black">Operazione binaria</font> (''It''). <font color="<ins class="diffchange diffchange-inline">purple</ins>"><ins class="diffchange diffchange-inline">二項演算</ins></font> (''<ins class="diffchange diffchange-inline">Ja</ins>''). <font color="<ins class="diffchange diffchange-inline">brown</ins>"><ins class="diffchange diffchange-inline">Бинарная операция</ins></font> (''<ins class="diffchange diffchange-inline">Ru</ins>''). <font color="green">Operación binaria</font> (''Sp'').</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
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BrianMcMahon
https://dictionary.iucr.org/index.php?title=Binary_operation&diff=4390&oldid=prev
BrianMcMahon: Tidied translations.
2017-11-09T16:00:40Z
<p>Tidied translations.</p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 16:00, 9 November 2017</td>
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<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><font color="blue">Loi de composition</font> (''Fr''). <font color="red">Zweistellige Verknüpfung</font> (''Ge''). <font color="<del class="diffchange diffchange-inline">green</del>"><del class="diffchange diffchange-inline">Operación Binaria</del></font> (''<del class="diffchange diffchange-inline">Sp</del>''). <font color="brown">Бинарная операция</font> (''Ru''). <font color="<del class="diffchange diffchange-inline">black</del>"><del class="diffchange diffchange-inline">Operazione binaria</del></font> (''<del class="diffchange diffchange-inline">It</del>''). <font color="<del class="diffchange diffchange-inline">purple</del>"><del class="diffchange diffchange-inline">二項演算</del></font> (''<del class="diffchange diffchange-inline">Ja</del>'').  </div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><font color="blue">Loi de composition</font> (''Fr''). <font color="red">Zweistellige Verknüpfung</font> (''Ge''). <font color="<ins class="diffchange diffchange-inline">black</ins>"><ins class="diffchange diffchange-inline">Operazione binaria</ins></font> (''<ins class="diffchange diffchange-inline">It</ins>''). <font color="brown">Бинарная операция</font> (''Ru''). <font color="<ins class="diffchange diffchange-inline">purple</ins>"><ins class="diffchange diffchange-inline">二項演算</ins></font> (''<ins class="diffchange diffchange-inline">Ja</ins>''). <font color="<ins class="diffchange diffchange-inline">green</ins>"><ins class="diffchange diffchange-inline">Operación binaria</ins></font> (''<ins class="diffchange diffchange-inline">Sp</ins>'').</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
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BrianMcMahon
https://dictionary.iucr.org/index.php?title=Binary_operation&diff=3929&oldid=prev
BrianMcMahon: Style edits to align with printed edition
2017-05-12T12:48:55Z
<p>Style edits to align with printed edition</p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 12:48, 12 May 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4" >Line 4:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>A '''binary operation''' on a set ''S'' is a [[mapping]] ''f'' from the [[Cartesian product]] ''S × S'' to ''S''. A mapping from ''K x S'' to ''S'', where ''K'' need not be ''S'', is called an '''external binary operation'''.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>A '''binary operation''' on a set ''S'' is a [[mapping]] ''f'' from the [[Cartesian product]] ''S × S'' to ''S''. A mapping from ''K x S'' to ''S'', where ''K'' need not be ''S'', is called an '''external binary operation'''.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Many binary operations are commutative <del class="diffchange diffchange-inline">(</del>i.e. ''f(a,b) = f(b,a)'' holds for all ''a, b'' in ''S''<del class="diffchange diffchange-inline">) </del>or associative <del class="diffchange diffchange-inline">(</del>i.e.  ''f(f(a,b), c) = f(a, f(b,c))'' holds for all ''a,b,c'' in ''S''<del class="diffchange diffchange-inline">)</del>. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions or symmetry operations.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Many binary operations are commutative <ins class="diffchange diffchange-inline">[''</ins>i.e.<ins class="diffchange diffchange-inline">'' </ins>''f(a,b) = f(b,a)'' holds for all ''a, b'' in ''S''<ins class="diffchange diffchange-inline">] </ins>or associative <ins class="diffchange diffchange-inline">[''</ins>i.e.<ins class="diffchange diffchange-inline">'' </ins> ''f(f(a,b), c) = f(a, f(b,c))'' holds for all ''a,b,c'' in ''S''<ins class="diffchange diffchange-inline">]</ins>. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions or symmetry operations.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Examples of binary operations that are not commutative are subtraction (-), division (/), exponentiation(^), super-exponentiation(@)<del class="diffchange diffchange-inline">, </del>and composition.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Examples of binary operations that are not commutative are subtraction (-), division (/), exponentiation(^), super-exponentiation(@) and composition.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Binary operations are often written using infix notation such as ''a * b'', ''a + b''<del class="diffchange diffchange-inline">, </del>or ''a · b'' rather than by functional notation of the form ''f(a,b)''. Sometimes they are even written just by concatenation: ''ab''.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Binary operations are often written using infix notation such as ''a * b'', ''a + b'' or ''a · b''<ins class="diffchange diffchange-inline">, </ins>rather than by functional notation of the form ''f(a,b)''. Sometimes they are even written just by concatenation: ''ab''.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Fundamental crystallography]]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Fundamental crystallography]]</div></td></tr>
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BrianMcMahon
https://dictionary.iucr.org/index.php?title=Binary_operation&diff=2966&oldid=prev
BerndSouvignier at 14:02, 1 April 2009
2009-04-01T14:02:21Z
<p></p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 14:02, 1 April 2009</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>A '''binary operation''' on a set S is a <del class="diffchange diffchange-inline">function </del>''f'' from the [[Cartesian product]] S × S to S. A <del class="diffchange diffchange-inline">binary function </del>from K <del class="diffchange diffchange-inline">and </del>S to S, where K need not be S, is called an ''' external binary operation '''.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>A '''binary operation''' on a set <ins class="diffchange diffchange-inline">''</ins>S<ins class="diffchange diffchange-inline">'' </ins>is a <ins class="diffchange diffchange-inline">[[mapping]] </ins>''f'' from the [[Cartesian product]] <ins class="diffchange diffchange-inline">''</ins>S × S<ins class="diffchange diffchange-inline">'' </ins>to <ins class="diffchange diffchange-inline">''</ins>S<ins class="diffchange diffchange-inline">''</ins>. A <ins class="diffchange diffchange-inline">mapping </ins>from <ins class="diffchange diffchange-inline">''</ins>K <ins class="diffchange diffchange-inline">x </ins>S<ins class="diffchange diffchange-inline">'' </ins>to <ins class="diffchange diffchange-inline">''</ins>S<ins class="diffchange diffchange-inline">''</ins>, where <ins class="diffchange diffchange-inline">''</ins>K<ins class="diffchange diffchange-inline">'' </ins>need not be <ins class="diffchange diffchange-inline">''</ins>S<ins class="diffchange diffchange-inline">''</ins>, is called an '''external binary operation'''.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Many binary operations are commutative or associative. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions <del class="diffchange diffchange-inline">on a single set</del>.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Many binary operations are commutative <ins class="diffchange diffchange-inline">(i.e. ''f(a,b) = f(b,a)'' holds for all ''a, b'' in ''S'') </ins>or associative <ins class="diffchange diffchange-inline">(i.e.  ''f(f(a,b), c) = f(a, f(b,c))'' holds for all ''a,b,c'' in ''S'')</ins>. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions <ins class="diffchange diffchange-inline">or symmetry operations</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Examples of operations that are not commutative are subtraction (-), division (/), exponentiation(^), <del class="diffchange diffchange-inline">and </del>super-exponentiation(@).</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Examples of <ins class="diffchange diffchange-inline">binary </ins>operations that are not commutative are subtraction (-), division (/), exponentiation(^), super-exponentiation(@)<ins class="diffchange diffchange-inline">, and composition</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Binary operations are often written using infix notation such as a * b, a + b, or a · b rather than by functional notation of the form ''f<del class="diffchange diffchange-inline">''</del>(a,b). Sometimes they are even written just by <del class="diffchange diffchange-inline">juxtaposition</del>: ab.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Binary operations are often written using infix notation such as <ins class="diffchange diffchange-inline">''</ins>a * b<ins class="diffchange diffchange-inline">''</ins>, <ins class="diffchange diffchange-inline">''</ins>a + b<ins class="diffchange diffchange-inline">''</ins>, or <ins class="diffchange diffchange-inline">''</ins>a · b<ins class="diffchange diffchange-inline">'' </ins>rather than by functional notation of the form ''f(a,b)<ins class="diffchange diffchange-inline">''</ins>. Sometimes they are even written just by <ins class="diffchange diffchange-inline">concatenation</ins>: <ins class="diffchange diffchange-inline">''</ins>ab<ins class="diffchange diffchange-inline">''</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Fundamental crystallography]]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Fundamental crystallography]]</div></td></tr>
</table>
BerndSouvignier
https://dictionary.iucr.org/index.php?title=Binary_operation&diff=2612&oldid=prev
MassimoNespolo at 09:16, 29 May 2007
2007-05-29T09:16:59Z
<p></p>
<p><b>New page</b></p><div><font color="blue">Loi de composition</font> (''Fr''). <font color="red">Zweistellige Verknüpfung</font> (''Ge''). <font color="green">Operación Binaria</font> (''Sp''). <font color="brown">Бинарная операция</font> (''Ru''). <font color="black">Operazione binaria</font> (''It''). <font color="purple">二項演算</font> (''Ja''). <br />
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A '''binary operation''' on a set S is a function ''f'' from the [[Cartesian product]] S × S to S. A binary function from K and S to S, where K need not be S, is called an ''' external binary operation '''.<br />
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Many binary operations are commutative or associative. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions on a single set.<br />
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Examples of operations that are not commutative are subtraction (-), division (/), exponentiation(^), and super-exponentiation(@).<br />
<br />
Binary operations are often written using infix notation such as a * b, a + b, or a · b rather than by functional notation of the form ''f''(a,b). Sometimes they are even written just by juxtaposition: ab.<br />
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[[Category:Fundamental crystallography]]</div>
MassimoNespolo