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Groupoid

A groupoid is a non-empty set  with a binary operation defined on it. (Distinguish this from the notion of groupoid introduced by Heinrich Brandt and used in the category theory and homotopy theory for  simultaneous generalization the notions of a  group, equivalence relations on sets, and actions of groups on sets).

For a groupoid also term ''magma'' is used, a term introduced by Bourbaki.

Groupoid with associative is called semigroup or associative groupoid. If the operation is commutative the groupoid is called commutative.

A free groupoid on a set X is the groupoid generated by the set X  in such a way that is there are no relations or axioms imposed on the generators.

If the operation  possesses the unit element, say , the groupoid is called a groupoid with unity (or with neutral) element. This means that there is an such that for each .

A groupoid in which each element is invertible is called quasigroup.

A groupoid is called a groupoid with cancellation if either of the equations and implies for all .  For a definition of a groupoid with right cancellation  and a groupoid with left cancellation  consult .

A groupoid is called a division groupoid (or groupoid with division) if the equations and are solvable in (not necessarily uniquely).

Groupoid which is simultaneously a groupoid with division and cancellation is a quasigroup.

Any groupoid with cancellation is imbeddable into a quasigroup.

A homomorphic image of a quasigroup is a groupoid with division.

Given a groupoid , we can define the so-called inverse groupoid by .

An important concept in the theory of groupoids is that of isotopy of operations. On a set  let two binary operations, say   and are defined.  These two operations are called isotropic if there exist three one-to-one mappings   of onto itself such that for all .

A groupoid that is isotopic to a quasigroup is itself a quasigroup.

Note that for groups the notions of the isotopy and isomorphism coincide. Namely, a groupoid with a unit element that is isotopic to a group, is also isomorphic to this group.

A groupoid is said to be left alternative if for all , and dually right alternative if for all . If   is both left and right alternative is said to be alternative. An associative groupoid (semigroup) is clearly alternative.

A groupoid in which every pair of elements generates an associative subgroupoid  must be alternative. In contrast to the so-called Artin’s theorem for algebras where these two statements are equivalent, the converse of the above implication is not true for groupoids.

A groupoid is said to be power associative if the subgroupoid generated by its any element is associative. This means that if an element is multiplied by itself several times, it does not matter in which order the multiplications are carried out. For instance , but also . Power associativity is stronger than merely requiring that for every .

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