Groupoid algebra
In mathematics, the concept of groupoid algebra generalizes the notion of group algebra. Definition 
Given a groupoid  (in the sense of a category with all arrows invertible) and a field , it is possible to define the groupoid algebra  as the algebra over  formed by the vector space having the elements of (the arrows of)  as generators and having the multiplication of these elements defined by , whenever this product is defined, and  otherwise. The product is then extended by linearity. Examples 
Some examples of groupoid algebras are the following:
 Group rings
 Matrix algebras
 Algebras of functions

Properties 
 When a groupoid has a finite number of objects and a finite number of morphisms, the groupoid algebra is a direct sum of tensor products of group algebras and matrix algebras. See also 
 Hopf algebra
 Partial group algebra

Notes

References 
 
 
 
 

Algebra