Actions

Difference between revisions of "Groupoid"

From Online Dictionary of Crystallography

 
m
Line 1: Line 1:
<font color="blue">Grupoïde</font> (''Fr''); <font color="red">Gruppoid</font> (''Ge''); <font color="green">Grupoide</font> (''Sp''); <font color="black">Gruppoide</font> (''It''); <font color="purple">亜群</font> (''Ja'').
+
<font color="blue">Groupoïde</font> (''Fr''); <font color="red">Gruppoid</font> (''Ge''); <font color="green">Grupoide</font> (''Sp''); <font color="black">Gruppoide</font> (''It''); <font color="purple">亜群</font> (''Ja'').
  
  

Revision as of 09:26, 28 February 2007

Groupoïde (Fr); Gruppoid (Ge); Grupoide (Sp); Gruppoide (It); 亜群 (Ja).


A groupoid (G,*) is a set G with a law of composition * mapping of a subset of G x 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 ex (left unit of x), ex' (right unit of x) and x-1 ("inverse" of x) such that:
    • ex*x = x
    • x* ex' = x
    • x-1*x = ex'.

From these properties it follows that:

  • x* x-1 = ex, i.e. that that ex is right unit for x-1,
  • ex' is left unit for x-1
  • ex and ex' are idempotents, i.e. ex* ex = ex and ex'* ex' = ex'.

See also

OD structure