Cartesian monoid explained

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.^[1]

Definition

\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 ( - , - ) .

Notes and References

1. .