Groupoid
From Online Dictionary of Crystallography
Groupoïde (Fr). Gruppoid (Ge). Gruppoide (It). 亜群 (Ja). Группоид (Ru). Grupoide (Sp).
A groupoid (G,*) is a set G with a law of composition * mapping of a subset of G × G into G. The properties of a groupoid are:
- if x, y, z ∈ G and if one of the compositions (x*y)*z or x*(y*z) is defined, so is the other and they are equal (associativity);
- if x, x' and y ∈ G are such that x*y and x' * y are defined and equal, then x = x' (cancellation property);
- for all x ∈ G there exist elements e_{x} (left unit of x), e_{x}' (right unit of x) and x^{−1} ('inverse' of x) such that:
- e_{x} * x = x
- x * e_{x}' = x
- x^{−1} * x = e_{x}'.
From these properties it follows that:
- x * x^{−1} = e_{x}, i.e. that e_{x} is right unit for x^{−1},
- e_{x}' is left unit for x^{−1}
- e_{x} and e_{x}' are idempotents, i.e. e_{x} * e_{x} = e_{x} and e_{x}' * e_{x}' = e_{x}'.
A groupoid can be seen as:
- a group with a partial function replacing the binary operation;
- a category in which every morphism is invertible.
The concept of groupoid as defined here was introduced by Brandt (1927). An alternative meaning of groupoid was introduced by Hausmann and Ore (1937) as a set on which binary operations act but neither the identity nor the inversion are included. For this second meaning nowadays the term magma is used instead (Bourbaki, 1998).
References
- Bourbaki, N. (1998). Elements of Mathematics: Algebra 1. Springer.
- Brandt, H. (1927). Mathematische Annalen, 96, 360-366.
- Hausmann, B. A. and Ore, O. (1937). American Journal of Mathematics, 59, 983-1004.