virtual group

There are at least three definitions of "groupoid" currently in use. The first type of groupoid is an algebraic structure on a set with a binary operator. The only restriction on the operator is closure (i.e., applying the binary operator to two elements of a given set S returns a value which is itself a member of S). Associativity, commutativity, etc., are not required. A groupoid can be empty. The numbers of nonisomorphic groupoids of this type having n = 1, 2, ... elements are 1, 10, 3330, 178981952, ... (OEIS A001329), and the corresponding numbers of nonisomorphic and nonantiisomorphic groupoids are 1, 7, 1734, 89521056, ... (OEIS A001424). An associative groupoid is called a semigroup.

binary operator | etale space | fundamental group | holonomy group | inverse semigroup | Lie algebra | Lie algebroid | Lie group | Lie groupoid | monoid | quasigroup | semigroup | stack of groupoids | topological groupoid