View source for Isotopy (in algebra)

''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>

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.

====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><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bruck,   "A survey of binary systems" , Springer  (1971)</TD></TR></table>