Difference between revisions of "Isotopy (in algebra)"
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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" />, | ||
Revision as of 07:04, 6 May 2012
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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) |
Comments
References
| [a1] | R.H. Bruck, "A survey of binary systems" , Springer (1971) |
Isotopy (in algebra). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isotopy_(in_algebra)&oldid=13308