Commutative Groupoid

We call ($A,1,*$), a Commutative Groupoid where

Sets

• $A$

Functions

• $1:()\to A$
• $*:A\times A\to A$

Properties

• Associative:
  $\forall$ $x\in A$, $y\in A$, $z\in A$,
  $(x*y)*z=x*(y*z)$
• Identity:
  $\forall$ $x\in A$,
  $x*1=x$
• Cancellative:
  $\forall$ $x\in A$, $y\in A$, $z\in A$,
  $z*x=z*y\to x=y\vee y*z=y*z\to x=y$
• Commutative:
  $\forall$ $x\in A$, $y\in A$,
  $x*y=y*x$
• Closure (constant):
  $1\in A$

Similar Structures

Stronger Forms

• The Commutative Groupoid can be extended into an Abelian Group
  

  Commutative Groupoid	Abelian Group
  $*:A\times A\to A$	$*:A\times A\to A$
  $1:()\to A$	$1:()\to A$
  $A$	$A$

  Additional Properties
  $*$
  • Closure

Weaker Forms

• The Commutative Groupoid can be reduced into a Groupoid
  

  Commutative Groupoid	Groupoid
  $*:A\times A\to A$	$*:A\times A\to A$
  $1:()\to A$	$1:()\to A$
  $A$	$A$

  Additional Properties
  $*$
  • Commutative