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Cartesian monoid

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Source updated August 1, 2023

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.

01Definition

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 $(-,-)$ .

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This article is adapted from the Wikipedia article “Cartesian monoid”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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