# Difference between revisions of "Groupoid"

A universal algebra with one binary operation. It is the broadest class of such algebras: groups, semi-groups, quasi-groups — all these are groupoids of a special type. An important concept in the theory of groupoids is that of isotopy of operations. On a set $G$ let there be defined two binary operations, denoted by $(\cdot)$ and $(\circ)$; they are isotopic if there exist three one-to-one mappings $\alpha$, $\beta$ and $\gamma$ of $G$ onto itself such that $a\cdot b=\gamma^{-1}(\alpha a\circ\beta b)$ for all $a,b\in G$. A groupoid that is isotopic to a quasi-group is itself a quasi-group; a groupoid with a unit element that is isotopic to a group, is also isomorphic to this group. For this reason, in group theory the concept of isotopy is not used: For groups isotopy and isomorphism coincide.
A groupoid with cancellation is a groupoid in which either of the equations $ab=ac$, $ba=ca$ implies $b=c$, where $a$, $b$ and $c$ are elements of the groupoid. Any groupoid with cancellation is imbeddable into a quasi-group. A homomorphic image of a quasi-group is a groupoid with division, that is, a groupoid in which the equations $ax=b$ and $ya=b$ are solvable (but do not necessarily have unique solutions).