https://groupprops.subwiki.org/w/index.php?action=history&feed=atom&title=Inverse_elementInverse element - Revision history2026-08-30T12:18:02ZRevision history for this page on the wikiMediaWiki 1.41.2https://groupprops.subwiki.org/w/index.php?title=Inverse_element&diff=6086&oldid=prevVipul: 1 revision2008-05-07T23:45:20Z<p>1 revision</p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<tr class="diff-title" lang="en">
<td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:45, 7 May 2008</td>
</tr><tr><td colspan="2" class="diff-notice" lang="en"><div class="mw-diff-empty">(No difference)</div>
</td></tr></table>Vipulhttps://groupprops.subwiki.org/w/index.php?title=Inverse_element&diff=6085&oldid=prevVipul at 12:12, 8 March 20072007-03-08T12:12:44Z<p></p>
<p><b>New page</b></p><div>''This is the default notion of inverse element. There is another, more general notion of inverse element in a [[semigroup]], which does not depend on existence of a [[neutral element]]''<br />
<br />
==Definition==<br />
<br />
===Definition with symbols===<br />
<br />
Given a set <math>S</math> with a binary operation <math>*</math> and a [[neutral element]] <math>e</math> for <math>*</math>, and given elements <math>a</math> and <math>b</math> we say that:<br />
<br />
* <math>b</math> is a '''left inverse''' to <math>a</math> if <math>b * a = e</math><br />
* <math>b</math> is a '''right inverse''' to <math>a</math> if <math>a * b = e</math><br />
* <math>b</math> is an '''inverse''' or '''two-sided inverse''' to <math>a</math> if <math>a * b = b * a = e</math> (that is, <math>b</math> is both a left and a right inverse to <math>a</math>)<br />
<br />
An element which possesses a (left/right) inverse is termed (left/right) [[invertible element|invertible]].<br />
<br />
==Facts==<br />
<br />
===Equality of left and right inverses===<br />
<br />
If <math>*</math> is an [[associative binary operation]], and an element has both a left and a right inverse with respect to <math>*</math>, then the left and right inverse are equal.<br />
<br />
To prove this, let <math>a</math> be an element of <math>S</math> with left inverse <math>b</math> and right inverse <math>c</math>. Then, <math>(b * a) * c = b * (a * c)</math> by associativity. The left side simplifies to <math>e * c = c</math> while the right side simplifies to <math>b * e = b</math>. Hence, <math>b = c</math>.<br />
<br />
Some easy corollaries:<br />
<br />
* If an element has a left inverse, it can have at most one right inverse; moreover, if the right inverse exists, it must be equal to the left inverse, and is thus a two-sided inverse<br />
* If an element has a right inverse, it can have at most one left inverse; moreover, if the left inverse exists, it must be equal to the right inverse, and is thus a two-sided inverse</div>Vipul