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.
A Cartesian monoid is a structure with signature
⟨
∗
,
e
,
(
−
,
−
)
,
L
,
R
⟩
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
(
−
,
−
)
.
