Cartesian monoid

A Cartesian monoid is a monoid, with additional structure of pairing and projection operators. It was first formulated by Dana Scott and Joachim Lambek independently.

Definition
A Cartesian monoid is a structure with signature $$\langle *,e,(-,-),L,R\rangle$$ where $$*$$ and $$(-,-)$$ are binary operations, $$L, R$$, and $$e$$ are constants satisfying the following axioms for all $$x,y,z$$ in its universe:
 * Monoid : $$*$$ is a monoid with identity $$e$$
 * Left Projection   : $$L * (x,\,y) = x $$
 * Right Projection  :$$R * (x,\,y) = y$$
 * Surjective Pairing :$$ (L*x,\,R*x) = x$$
 * Right Homogeneity :$$ (x*z,\,y*z)=(x,\,y) * z$$

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