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Difference between revisions of "Isotopy (in algebra)"

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''For other meanings of the term see the disambiguation page [[Isotopy]]''.
  
A relation on the class of all groupoids (cf. [[Groupoid|Groupoid]]) defined on a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i0529301.png" />. Namely, two groupoids on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i0529302.png" /> are called isotopic if there exist permutations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i0529303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i0529304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i0529305.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i0529306.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i0529307.png" />,
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i0529308.png" /></td> </tr></table>
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A relation on the class of all [[magma]]s defined by [[binary operation]]s on a given set $M$. Namely, two operations $(M,{\cdot})$ and $(M,{\circ})$ are called isotopic if there exist permutations $\rho$, $\sigma$ and $\tau$ of $M$ such that for any $a,b\in M$,
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$$
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a\circ b=(a\rho\cdot b\sigma)\tau \ .
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i0529309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i05293010.png" /> denote the operations in these two groupoids. The isotopy relation is an equivalence relation for the binary operations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i05293011.png" />. An isomorphism of two binary operations defined on the same set is a special case of an isotopy (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i05293012.png" />). An isotopy is called principal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052930/i05293013.png" /> is the identity permutation. Every isotope (i.e. isotopic groupoid) of a groupoid is isomorphic to a principal isotope of the groupoid. Every groupoid that is isotopic to a [[Quasi-group|quasi-group]] is itself a quasi-group. Every quasi-group is isotopic to some [[Loop|loop]] (Albert's theorem). If a loop (in particular, a group) is isotopic to some group, then they are isomorphic. If a groupoid with identity is isotopic to a [[Semi-group|semi-group]], then they are isomorphic, that is, they are both semi-groups with identity.
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The isotopy relation is an equivalence relation for the binary operations on $M$. An isomorphism of two binary operations defined on the same set is a special case of an isotopy (with $\rho=\sigma=\tau^{-1}$). An isotopy is called ''principal'' if $\tau$ is the identity permutation. Every isotope (i.e. isotopic magma) of a magma is isomorphic to a principal isotope of the magma. Every magma that is isotopic to a [[quasi-group]] is itself a quasi-group. Every quasi-group is isotopic to some [[loop]] (Albert's theorem). If a loop (in particular, a group) is isotopic to some group, then they are isomorphic. If a magma with identity is isotopic to a [[semi-group]], then they are isomorphic, that is, they are both semi-groups with identity.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR>
====Comments====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bruck,  "A survey of binary systems" , Springer  (1971)</TD></TR>
 
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</table>
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bruck,  "A survey of binary systems" , Springer  (1971)</TD></TR></table>
 

Latest revision as of 19:21, 7 January 2016

For other meanings of the term see the disambiguation page Isotopy.


A relation on the class of all magmas defined by binary operations on a given set $M$. Namely, two operations $(M,{\cdot})$ and $(M,{\circ})$ are called isotopic if there exist permutations $\rho$, $\sigma$ and $\tau$ of $M$ such that for any $a,b\in M$, $$ a\circ b=(a\rho\cdot b\sigma)\tau \ . $$

The isotopy relation is an equivalence relation for the binary operations on $M$. An isomorphism of two binary operations defined on the same set is a special case of an isotopy (with $\rho=\sigma=\tau^{-1}$). An isotopy is called principal if $\tau$ is the identity permutation. Every isotope (i.e. isotopic magma) of a magma is isomorphic to a principal isotope of the magma. Every magma that is isotopic to a quasi-group is itself a quasi-group. Every quasi-group is isotopic to some loop (Albert's theorem). If a loop (in particular, a group) is isotopic to some group, then they are isomorphic. If a magma with identity is isotopic to a semi-group, then they are isomorphic, that is, they are both semi-groups with identity.

References

[1] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
[a1] R.H. Bruck, "A survey of binary systems" , Springer (1971)
How to Cite This Entry:
Isotopy (in algebra). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isotopy_(in_algebra)&oldid=26098