A Cartesian monoid is a

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...

, with additional structure of pairing and projection operators. It was first formulated by

Dana Scott Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, Ca ...

and

Joachim Lambek Joachim "Jim" Lambek (5 December 1922 – 23 June 2014) was a German-born Canadian mathematician. He was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his PhD degree in 1950 with Hans Zassenhaus a ...

independently..

Definition

A Cartesian monoid is a structure with

signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...

\langle *,e,(-,-),L,R\rangle

where

*

and

(-,-)

are

binary operations In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...

L, R

, and

e

are constants satisfying the following

axioms An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

for all

x,y,z

in its

universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the 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

{{reflist Mathematical logic