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 where and are binary operations, , and are constants satisfying the following axioms for all in its universe: Monoid is a monoid with identity Left Projection Right Projection Surjective Pairing Right Homogeneity
The interpretation is that and are left and right projection functions respectively for the pairing function .
References