User:KBlott/Definition/Commutative Class Monoid

Definition
Let $(C,*)$ be a class monoid. Then,
 * $(C,*)$ is called a commutative class monoid (or CC monoid)

iff
 * $\forall x,y \in C, x*y = y*x$

iff
 * $\forall x,y \in C,\left[ =(*(x,y),*(y,x))\right]$

iff
 * $\left[ x,y \in C \right] \implies \left[ =(*(x,y),*(y,x))\right]$

iff
 * $\left[ (x \in C)\wedge (y \in C) \right] \implies \left[ =(*(x,y),*(y,x))\right]$

iff
 * $\left[ (x \in C)\wedge (y \in C) \right] \implies \left[ (* (x,y)\subset *(y,x))\wedge (y,x)\subset *(x,y)) \right]$