Cartesian monoid

A Cartesian monoid is a structure with signature ${\displaystyle \langle *,e,(-,-),L,R\rangle }$  where ${\displaystyle *}$  and ${\displaystyle (-,-)}$  are binary operations, ${\displaystyle L,R}$ , and ${\displaystyle e}$  are constants satisfying the following axioms for all ${\displaystyle x,y,z}$  in its universe:

Monoid

${\displaystyle *}$  is a monoid with identity ${\displaystyle e}$ 

Left Projection

${\displaystyle L*(x,\,y)=x}$ 

Right Projection

${\displaystyle R*(x,\,y)=y}$ 

Surjective Pairing

${\displaystyle (L*x,\,R*x)=x}$ 

Right Homogeneity

${\displaystyle (x*z,\,y*z)=(x,\,y)*z}$ 

The interpretation is that ${\displaystyle L}$  and ${\displaystyle R}$  are left and right projection functions respectively for the pairing function ${\displaystyle (-,-)}$ .