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.<ref>{{citation | last = Statman | first = Rick | authorlink=Richard Statman | contribution = On Cartesian monoids | doi = 10.1007/3-540-63172-0_55 | mr = 1611514 | pages = 446–459 | publisher = Springer | location = Berlin | series = Lecture Notes in Computer Science | title = Computer science logic (Utrecht, 1996) | volume = 1258 | year = 1997}}.</ref>
== Definition ==
A Cartesian monoid is a structure with signature <math>\langle *,e,(-,-),L,R\rangle</math> where <math>*</math> and <math>(-,-)</math> are binary operations, <math>L, R</math>, and <math>e</math> are constants satisfying the following axioms for all <math>x,y,z</math> in its universe: ; '''Monoid''' : <math>*</math> is a monoid with identity <math>e</math> ; '''Left Projection''' : <math>L * (x,\,y) = x </math> ; '''Right Projection''' :<math>R * (x,\,y) = y</math> ; '''Surjective Pairing''' :<math> (L*x,\,R*x) = x</math> ; '''Right Homogeneity''' :<math> (x*z,\,y*z)=(x,\,y) * z</math>
The interpretation is that <math>L</math> and <math>R</math> are left and right projection functions respectively for the pairing function <math>(-,-)</math>.
==References== {{reflist}}
Category:Mathematical logic