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| | 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. |
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| | == Definition == |
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| | A Cartesian monoid is a [[Structure (mathematical logic)|structure]] with [[Signature (logic)|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 [[Domain of discourse|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> |
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| | The interpretation is that <math>L</math> and <math>R</math> are left and right projection functions respectively for the pairing function <math>(-,-)</math>. |
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| | [[Category:Mathematical logic]] |
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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.
Definition
A Cartesian monoid is a structure with signature where and are binary operations, , and are constants satisfying the following axioms for all in its universe:
- Monoid
- is a monoid with identity
- Left Projection
- Right Projection
- Surjective Pairing
- Right Homogeneity
The interpretation is that and are left and right projection functions respectively for the pairing function .