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}'.
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.