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 $(-,-)$.