What is the product of commutative and anticommutative on the category of free algebras?
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Let's say we have the category of free algebras. This is a category where objects are algebras defined in terms of identities. The morphisms of the category are sets of identities to be added. There is a morphism from $A$ to $B$ if there is a set of identities that can be added to $A$ to make an algebra equivalent to $B$.
Let's also have the commutative algebra to be an algebra with a binary operator $times$ such that:
$$atimes b=btimes a$$
and the anticommutative algebra to be the algebra with the operators $times$ and $-$ such that:
beginalign*
atimes (btimes -b) = a \
atimes (-btimes b) = a \
(-btimes b)times a = a \
(btimes -b)times a = a \
atimes b= -(btimes a)
endalign*
I have been looking for the product of these two algebras on the category but I haven't been able to find it. I think it is a magma but I haven't been able to prove it.
I did find that the coproduct of the two is the algebra:
beginalign*
a times b &= btimes a \
atimes (btimes -b) &= a \
a &= -a
endalign*
What is the product of the two algebras? Is it the magma? how can I prove it?
abstract-algebra category-theory
asked Aug 27 at 0:00
W W
1,000526
add a comment |Â
up vote
0
down vote
favorite
Let's say we have the category of free algebras. This is a category where objects are algebras defined in terms of identities. The morphisms of the category are sets of identities to be added. There is a morphism from $A$ to $B$ if there is a set of identities that can be added to $A$ to make an algebra equivalent to $B$.
Let's also have the commutative algebra to be an algebra with a binary operator $times$ such that:
$$atimes b=btimes a$$
and the anticommutative algebra to be the algebra with the operators $times$ and $-$ such that:
beginalign*
atimes (btimes -b) = a \
atimes (-btimes b) = a \
(-btimes b)times a = a \
(btimes -b)times a = a \
atimes b= -(btimes a)
endalign*
I have been looking for the product of these two algebras on the category but I haven't been able to find it. I think it is a magma but I haven't been able to prove it.
I did find that the coproduct of the two is the algebra:
beginalign*
a times b &= btimes a \
atimes (btimes -b) &= a \
a &= -a
endalign*
What is the product of the two algebras? Is it the magma? how can I prove it?
abstract-algebra category-theory
asked Aug 27 at 0:00
W W
1,000526
Can you give a precise definition for this category?
– Eric Wofsey
Aug 27 at 0:36
@EricWofsey I've expanded a bit but the definition seems pretty clear to me. If there is any specific point of confusion I'd be happy to address that.
– W W
Aug 27 at 3:36
The definition is very far from clear. What does it mean that "objects are algebras defined in terms of identities"? Is an object a single algebra, i.e. a set equipped with some operations? Or is an object a variety of algebras, in the sense of universal algebra? What do your morphisms mean if the operations on $A$ and $B$ are not the same? Is your category a preorder, or can different sets of identities correspond to different morphisms between the same two objects? What do you mean by "an algebra equivalent to $B$"?
– Eric Wofsey
Aug 27 at 3:50
What is the signature of your algebras?
– J.-E. Pin
Aug 27 at 3:53
@J.-E.Pin For the purpose of this question we can just say $langle S, +, -rangle$ where $+$ is binary and $-$ is unary because those are the only operations my problem cares about. In actuality it would be equipt with an infinite number of operations both of every airity. Without identities to contrain them additional operators make no difference so we can add as many as we would like.
– W W
Aug 27 at 3:57
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let's say we have the category of free algebras. This is a category where objects are algebras defined in terms of identities. The morphisms of the category are sets of identities to be added. There is a morphism from $A$ to $B$ if there is a set of identities that can be added to $A$ to make an algebra equivalent to $B$.
Let's also have the commutative algebra to be an algebra with a binary operator $times$ such that:
$$atimes b=btimes a$$
and the anticommutative algebra to be the algebra with the operators $times$ and $-$ such that:
beginalign*
atimes (btimes -b) = a \
atimes (-btimes b) = a \
(-btimes b)times a = a \
(btimes -b)times a = a \
atimes b= -(btimes a)
endalign*
I have been looking for the product of these two algebras on the category but I haven't been able to find it. I think it is a magma but I haven't been able to prove it.
I did find that the coproduct of the two is the algebra:
beginalign*
a times b &= btimes a \
atimes (btimes -b) &= a \
a &= -a
endalign*
What is the product of the two algebras? Is it the magma? how can I prove it?
abstract-algebra category-theory
asked Aug 27 at 0:00
W W
1,000526
Let's say we have the category of free algebras. This is a category where objects are algebras defined in terms of identities. The morphisms of the category are sets of identities to be added. There is a morphism from $A$ to $B$ if there is a set of identities that can be added to $A$ to make an algebra equivalent to $B$.
Let's also have the commutative algebra to be an algebra with a binary operator $times$ such that:
$$atimes b=btimes a$$
and the anticommutative algebra to be the algebra with the operators $times$ and $-$ such that:
beginalign*
atimes (btimes -b) = a \
atimes (-btimes b) = a \
(-btimes b)times a = a \
(btimes -b)times a = a \
atimes b= -(btimes a)
endalign*
I have been looking for the product of these two algebras on the category but I haven't been able to find it. I think it is a magma but I haven't been able to prove it.
I did find that the coproduct of the two is the algebra:
beginalign*
a times b &= btimes a \
atimes (btimes -b) &= a \
a &= -a
endalign*
What is the product of the two algebras? Is it the magma? how can I prove it?
abstract-algebra category-theory
asked Aug 27 at 0:00
W W
1,000526
edited Aug 27 at 3:35
asked Aug 27 at 0:00
W W
1,000526
asked Aug 27 at 0:00
W W
1,000526
asked Aug 27 at 0:00
W W
1,000526
1,000526
Can you give a precise definition for this category?
– Eric Wofsey
Aug 27 at 0:36
@EricWofsey I've expanded a bit but the definition seems pretty clear to me. If there is any specific point of confusion I'd be happy to address that.
– W W
Aug 27 at 3:36
The definition is very far from clear. What does it mean that "objects are algebras defined in terms of identities"? Is an object a single algebra, i.e. a set equipped with some operations? Or is an object a variety of algebras, in the sense of universal algebra? What do your morphisms mean if the operations on $A$ and $B$ are not the same? Is your category a preorder, or can different sets of identities correspond to different morphisms between the same two objects? What do you mean by "an algebra equivalent to $B$"?
– Eric Wofsey
Aug 27 at 3:50
What is the signature of your algebras?
– J.-E. Pin
Aug 27 at 3:53
@J.-E.Pin For the purpose of this question we can just say $langle S, +, -rangle$ where $+$ is binary and $-$ is unary because those are the only operations my problem cares about. In actuality it would be equipt with an infinite number of operations both of every airity. Without identities to contrain them additional operators make no difference so we can add as many as we would like.
– W W
Aug 27 at 3:57
add a comment |Â
Can you give a precise definition for this category?
– Eric Wofsey
Aug 27 at 0:36
@EricWofsey I've expanded a bit but the definition seems pretty clear to me. If there is any specific point of confusion I'd be happy to address that.
– W W
Aug 27 at 3:36
The definition is very far from clear. What does it mean that "objects are algebras defined in terms of identities"? Is an object a single algebra, i.e. a set equipped with some operations? Or is an object a variety of algebras, in the sense of universal algebra? What do your morphisms mean if the operations on $A$ and $B$ are not the same? Is your category a preorder, or can different sets of identities correspond to different morphisms between the same two objects? What do you mean by "an algebra equivalent to $B$"?
– Eric Wofsey
Aug 27 at 3:50
What is the signature of your algebras?
– J.-E. Pin
Aug 27 at 3:53
@J.-E.Pin For the purpose of this question we can just say $langle S, +, -rangle$ where $+$ is binary and $-$ is unary because those are the only operations my problem cares about. In actuality it would be equipt with an infinite number of operations both of every airity. Without identities to contrain them additional operators make no difference so we can add as many as we would like.
– W W
Aug 27 at 3:57
Can you give a precise definition for this category?
– Eric Wofsey
Aug 27 at 0:36
Can you give a precise definition for this category?
– Eric Wofsey
Aug 27 at 0:36
@EricWofsey I've expanded a bit but the definition seems pretty clear to me. If there is any specific point of confusion I'd be happy to address that.
– W W
Aug 27 at 3:36
@EricWofsey I've expanded a bit but the definition seems pretty clear to me. If there is any specific point of confusion I'd be happy to address that.
– W W
Aug 27 at 3:36
The definition is very far from clear. What does it mean that "objects are algebras defined in terms of identities"? Is an object a single algebra, i.e. a set equipped with some operations? Or is an object a variety of algebras, in the sense of universal algebra? What do your morphisms mean if the operations on $A$ and $B$ are not the same? Is your category a preorder, or can different sets of identities correspond to different morphisms between the same two objects? What do you mean by "an algebra equivalent to $B$"?
– Eric Wofsey
Aug 27 at 3:50
The definition is very far from clear. What does it mean that "objects are algebras defined in terms of identities"? Is an object a single algebra, i.e. a set equipped with some operations? Or is an object a variety of algebras, in the sense of universal algebra? What do your morphisms mean if the operations on $A$ and $B$ are not the same? Is your category a preorder, or can different sets of identities correspond to different morphisms between the same two objects? What do you mean by "an algebra equivalent to $B$"?
– Eric Wofsey
Aug 27 at 3:50
What is the signature of your algebras?
– J.-E. Pin
Aug 27 at 3:53
What is the signature of your algebras?
– J.-E. Pin
Aug 27 at 3:53
@J.-E.Pin For the purpose of this question we can just say $langle S, +, -rangle$ where $+$ is binary and $-$ is unary because those are the only operations my problem cares about. In actuality it would be equipt with an infinite number of operations both of every airity. Without identities to contrain them additional operators make no difference so we can add as many as we would like.
– W W
Aug 27 at 3:57
@J.-E.Pin For the purpose of this question we can just say $langle S, +, -rangle$ where $+$ is binary and $-$ is unary because those are the only operations my problem cares about. In actuality it would be equipt with an infinite number of operations both of every airity. Without identities to contrain them additional operators make no difference so we can add as many as we would like.
– W W
Aug 27 at 3:57
add a comment |Â
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Can you give a precise definition for this category?
– Eric Wofsey
Aug 27 at 0:36
@EricWofsey I've expanded a bit but the definition seems pretty clear to me. If there is any specific point of confusion I'd be happy to address that.
– W W
Aug 27 at 3:36
The definition is very far from clear. What does it mean that "objects are algebras defined in terms of identities"? Is an object a single algebra, i.e. a set equipped with some operations? Or is an object a variety of algebras, in the sense of universal algebra? What do your morphisms mean if the operations on $A$ and $B$ are not the same? Is your category a preorder, or can different sets of identities correspond to different morphisms between the same two objects? What do you mean by "an algebra equivalent to $B$"?
– Eric Wofsey
Aug 27 at 3:50
What is the signature of your algebras?
– J.-E. Pin
Aug 27 at 3:53
@J.-E.Pin For the purpose of this question we can just say $langle S, +, -rangle$ where $+$ is binary and $-$ is unary because those are the only operations my problem cares about. In actuality it would be equipt with an infinite number of operations both of every airity. Without identities to contrain them additional operators make no difference so we can add as many as we would like.
– W W
Aug 27 at 3:57