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.^[1]

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 $(-,-)$ .

References

^ Statman, Rick (1997), "On Cartesian monoids", Computer science logic (Utrecht, 1996), Lecture Notes in Comput. Sci., vol. 1258, Berlin: Springer, pp. 446–459, doi:10.1007/3-540-63172-0_55, MR 1611514.

[1] Statman, Rick (1997), "On Cartesian monoids", Computer science logic (Utrecht, 1996), Lecture Notes in Comput. Sci., vol. 1258, Berlin: Springer, pp. 446–459, doi:10.1007/3-540-63172-0_55, MR 1611514.

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