Monoid, group etc structures on M × N
#
In this file we define one-binop (Monoid
, Group
etc) structures on M × N
.
We also prove trivial simp
lemmas, and define the following operations on MonoidHom
s:
fst M N : M × N →* M
,snd M N : M × N →* N
: projectionsProd.fst
andProd.snd
asMonoidHom
s;inl M N : M →* M × N
,inr M N : N →* M × N
: inclusions of first/second monoid into the product;f.prod g
:M →* N × P
: sendsx
to(f x, g x)
;- When
P
is commutative,f.coprod g : M × N →* P
sends(x, y)
tof x * g y
(without the commutativity assumption onP
, seeMonoidHom.noncommPiCoprod
); f.prodMap g : M × N → M' × N'
:Prod.map f g
as aMonoidHom
, sends(x, y)
to(f x, g y)
.
Main declarations #
mulMulHom
/mulMonoidHom
: Multiplication bundled as a multiplicative/monoid homomorphism.divMonoidHom
: Division bundled as a monoid homomorphism.
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Coproduct of two MulHom
s with the same codomain:
f.coprod g (p : M × N) = f p.1 * g p.2
.
(Commutative codomain; for the general case, see MulHom.noncommCoprod
)
Equations
Instances For
Coproduct of two AddHom
s with the same codomain:
f.coprod g (p : M × N) = f p.1 + g p.2
.
(Commutative codomain; for the general case, see AddHom.noncommCoprod
)
Equations
Instances For
Given monoids M
, N
, the natural projection homomorphism from M × N
to M
.
Equations
Instances For
Given additive monoids A
, B
, the natural projection homomorphism
from A × B
to A
Equations
Instances For
Given monoids M
, N
, the natural projection homomorphism from M × N
to N
.
Equations
Instances For
Given additive monoids A
, B
, the natural projection homomorphism
from A × B
to B
Equations
Instances For
Given monoids M
, N
, the natural inclusion homomorphism from M
to M × N
.
Equations
Instances For
Given additive monoids A
, B
, the natural inclusion homomorphism
from A
to A × B
.
Equations
Instances For
Given monoids M
, N
, the natural inclusion homomorphism from N
to M × N
.
Equations
Instances For
Given additive monoids A
, B
, the natural inclusion homomorphism
from B
to A × B
.
Equations
Instances For
Combine two MonoidHom
s f : M →* N
, g : M →* P
into f.prod g : M →* N × P
given by (f.prod g) x = (f x, g x)
.
Equations
Instances For
Combine two AddMonoidHom
s f : M →+ N
, g : M →+ P
into
f.prod g : M →+ N × P
given by (f.prod g) x = (f x, g x)
Equations
Instances For
Equations
Instances For
Prod.map
as an AddMonoidHom
.
Equations
Instances For
Coproduct of two MonoidHom
s with the same codomain:
f.coprod g (p : M × N) = f p.1 * g p.2
.
(Commutative case; for the general case, see MonoidHom.noncommCoprod
.)
Equations
Instances For
Coproduct of two AddMonoidHom
s with the same codomain:
f.coprod g (p : M × N) = f p.1 + g p.2
.
(Commutative case; for the general case, see AddHom.noncommCoprod
.)
Equations
Instances For
The equivalence between M × N
and N × M
given by swapping the components
is multiplicative.
Equations
Instances For
The equivalence between M × N
and N × M
given by swapping the
components is additive.
Equations
Instances For
The equivalence between (M × N) × P
and M × (N × P)
is multiplicative.
Equations
Instances For
The equivalence between (M × N) × P
and M × (N × P)
is additive.
Equations
Instances For
Four-way commutativity of Prod
. The name matches mul_mul_mul_comm
.
Equations
Instances For
Four-way commutativity of Prod
.
The name matches mul_mul_mul_comm
Equations
Instances For
Product of multiplicative isomorphisms; the maps come from Equiv.prodCongr
.
Equations
Instances For
Product of additive isomorphisms; the maps come from Equiv.prodCongr
.
Equations
Instances For
Multiplying by the trivial monoid doesn't change the structure.
This is the MulEquiv
version of Equiv.uniqueProd
.
Equations
Instances For
Multiplying by the trivial monoid doesn't change the structure.
This is the AddEquiv
version of Equiv.uniqueProd
.
Equations
Instances For
Multiplying by the trivial monoid doesn't change the structure.
This is the MulEquiv
version of Equiv.prodUnique
.
Equations
Instances For
Multiplying by the trivial monoid doesn't change the structure.
This is the AddEquiv
version of Equiv.prodUnique
.
Equations
Instances For
The additive monoid equivalence between additive units of a product of two additive monoids, and the product of the additive units of each additive monoid.
Equations
Instances For
Multiplication and division as homomorphisms #
Multiplication as a monoid homomorphism.
Equations
Instances For
Addition as an additive monoid homomorphism.
Equations
Instances For
Division as a monoid homomorphism.
Equations
Instances For
Subtraction as an additive monoid homomorphism.