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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]]''.
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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" />,
 
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" />,
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====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><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bruck,  "A survey of binary systems" , Springer  (1971)</TD></TR></table>
 
 
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bruck,  "A survey of binary systems" , Springer  (1971)</TD></TR></table>
 

Revision as of 05:47, 7 May 2012

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


A relation on the class of all groupoids (cf. Groupoid) defined on a given set . Namely, two groupoids on are called isotopic if there exist permutations , and of such that for any ,

where and denote the operations in these two groupoids. The isotopy relation is an equivalence relation for the binary operations on . An isomorphism of two binary operations defined on the same set is a special case of an isotopy (with ). An isotopy is called principal if 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 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 groupoid 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=26140