Mixing
algebraic structure consisting of two fields with common multiplication operation
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I am looking for properties of an algebraic structure on a set defined as follows:
- two fields with distinct operations of addition (thus two distinct additive zeros) and a common operation of multiplication.
An example would be the set $mathbbR$ of real numbers and two operations of addition defined on it. For $a,bin mathbbR$:
- one is the usual addition $a+b$;
- the other is $adot+b=frac1frac1a+frac1b$.
The inverse of $ain mathbbR$ under both operations is $-a$. The additive zero of $+$ operation is the $0$, the additive zero of $dot+$ is $infty$ if we allow $infty=frac10$.
Any mention, examples, thoughts would be very welcome.
abstract-algebra
asked Jul 26 at 12:10
TooOldToLearn
997
 |Â
show 16 more comments
up vote
0
down vote
favorite
I am looking for properties of an algebraic structure on a set defined as follows:
- two fields with distinct operations of addition (thus two distinct additive zeros) and a common operation of multiplication.
An example would be the set $mathbbR$ of real numbers and two operations of addition defined on it. For $a,bin mathbbR$:
- one is the usual addition $a+b$;
- the other is $adot+b=frac1frac1a+frac1b$.
The inverse of $ain mathbbR$ under both operations is $-a$. The additive zero of $+$ operation is the $0$, the additive zero of $dot+$ is $infty$ if we allow $infty=frac10$.
Any mention, examples, thoughts would be very welcome.
abstract-algebra
asked Jul 26 at 12:10
TooOldToLearn
997
3
This is not clear. Can you give a detailed example of the structure you have in mind?
– lulu
Jul 26 at 12:13
4
In particular, how could there be two different additive zeroes? Since $0_itimes x=0_i$ for all $x$ in the field we see that $0_1times 0_2=0_2times 0_1$ is both $0_1$ and $0_2$.
– lulu
Jul 26 at 12:20
4
Not following. As a general rule, it's a bad idea to talk about algebraic structures with no examples in mind. The formalism is meant to generalize the properties you observe in particular examples. Here, I don't think you have any actual examples in mind. If you do, please edit your post to describe one or two examples in detail.
– lulu
Jul 26 at 12:36
2
Again, not following. If you have a precise example, please edit your post to include a detailed discussion of it.
– lulu
Jul 26 at 12:49
1
In the example you don't have two field structures on the same set. The joint set is $S=BbbRcupinfty$. You have one field structure on the set $Ssetminusinfty$ and another on $Ssetminus0$. They happen to share the multiplicative group, but the zero of one is outside the other field altogether.
– Jyrki Lahtonen
Jul 26 at 20:24
 |Â
show 16 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am looking for properties of an algebraic structure on a set defined as follows:
- two fields with distinct operations of addition (thus two distinct additive zeros) and a common operation of multiplication.
An example would be the set $mathbbR$ of real numbers and two operations of addition defined on it. For $a,bin mathbbR$:
- one is the usual addition $a+b$;
- the other is $adot+b=frac1frac1a+frac1b$.
The inverse of $ain mathbbR$ under both operations is $-a$. The additive zero of $+$ operation is the $0$, the additive zero of $dot+$ is $infty$ if we allow $infty=frac10$.
Any mention, examples, thoughts would be very welcome.
abstract-algebra
asked Jul 26 at 12:10
TooOldToLearn
997
I am looking for properties of an algebraic structure on a set defined as follows:
- two fields with distinct operations of addition (thus two distinct additive zeros) and a common operation of multiplication.
An example would be the set $mathbbR$ of real numbers and two operations of addition defined on it. For $a,bin mathbbR$:
- one is the usual addition $a+b$;
- the other is $adot+b=frac1frac1a+frac1b$.
The inverse of $ain mathbbR$ under both operations is $-a$. The additive zero of $+$ operation is the $0$, the additive zero of $dot+$ is $infty$ if we allow $infty=frac10$.
Any mention, examples, thoughts would be very welcome.
abstract-algebra
asked Jul 26 at 12:10
TooOldToLearn
997
edited Jul 26 at 20:35
asked Jul 26 at 12:10
TooOldToLearn
997
asked Jul 26 at 12:10
TooOldToLearn
997
asked Jul 26 at 12:10
TooOldToLearn
997
997
3
This is not clear. Can you give a detailed example of the structure you have in mind?
– lulu
Jul 26 at 12:13
4
In particular, how could there be two different additive zeroes? Since $0_itimes x=0_i$ for all $x$ in the field we see that $0_1times 0_2=0_2times 0_1$ is both $0_1$ and $0_2$.
– lulu
Jul 26 at 12:20
4
Not following. As a general rule, it's a bad idea to talk about algebraic structures with no examples in mind. The formalism is meant to generalize the properties you observe in particular examples. Here, I don't think you have any actual examples in mind. If you do, please edit your post to describe one or two examples in detail.
– lulu
Jul 26 at 12:36
2
Again, not following. If you have a precise example, please edit your post to include a detailed discussion of it.
– lulu
Jul 26 at 12:49
1
In the example you don't have two field structures on the same set. The joint set is $S=BbbRcupinfty$. You have one field structure on the set $Ssetminusinfty$ and another on $Ssetminus0$. They happen to share the multiplicative group, but the zero of one is outside the other field altogether.
– Jyrki Lahtonen
Jul 26 at 20:24
 |Â
show 16 more comments
3
This is not clear. Can you give a detailed example of the structure you have in mind?
– lulu
Jul 26 at 12:13
4
In particular, how could there be two different additive zeroes? Since $0_itimes x=0_i$ for all $x$ in the field we see that $0_1times 0_2=0_2times 0_1$ is both $0_1$ and $0_2$.
– lulu
Jul 26 at 12:20
4
Not following. As a general rule, it's a bad idea to talk about algebraic structures with no examples in mind. The formalism is meant to generalize the properties you observe in particular examples. Here, I don't think you have any actual examples in mind. If you do, please edit your post to describe one or two examples in detail.
– lulu
Jul 26 at 12:36
2
Again, not following. If you have a precise example, please edit your post to include a detailed discussion of it.
– lulu
Jul 26 at 12:49
1
In the example you don't have two field structures on the same set. The joint set is $S=BbbRcupinfty$. You have one field structure on the set $Ssetminusinfty$ and another on $Ssetminus0$. They happen to share the multiplicative group, but the zero of one is outside the other field altogether.
– Jyrki Lahtonen
Jul 26 at 20:24
3
3
This is not clear. Can you give a detailed example of the structure you have in mind?
– lulu
Jul 26 at 12:13
This is not clear. Can you give a detailed example of the structure you have in mind?
– lulu
Jul 26 at 12:13
4
4
In particular, how could there be two different additive zeroes? Since $0_itimes x=0_i$ for all $x$ in the field we see that $0_1times 0_2=0_2times 0_1$ is both $0_1$ and $0_2$.
– lulu
Jul 26 at 12:20
In particular, how could there be two different additive zeroes? Since $0_itimes x=0_i$ for all $x$ in the field we see that $0_1times 0_2=0_2times 0_1$ is both $0_1$ and $0_2$.
– lulu
Jul 26 at 12:20
4
4
Not following. As a general rule, it's a bad idea to talk about algebraic structures with no examples in mind. The formalism is meant to generalize the properties you observe in particular examples. Here, I don't think you have any actual examples in mind. If you do, please edit your post to describe one or two examples in detail.
– lulu
Jul 26 at 12:36
Not following. As a general rule, it's a bad idea to talk about algebraic structures with no examples in mind. The formalism is meant to generalize the properties you observe in particular examples. Here, I don't think you have any actual examples in mind. If you do, please edit your post to describe one or two examples in detail.
– lulu
Jul 26 at 12:36
2
2
Again, not following. If you have a precise example, please edit your post to include a detailed discussion of it.
– lulu
Jul 26 at 12:49
Again, not following. If you have a precise example, please edit your post to include a detailed discussion of it.
– lulu
Jul 26 at 12:49
1
1
In the example you don't have two field structures on the same set. The joint set is $S=BbbRcupinfty$. You have one field structure on the set $Ssetminusinfty$ and another on $Ssetminus0$. They happen to share the multiplicative group, but the zero of one is outside the other field altogether.
– Jyrki Lahtonen
Jul 26 at 20:24
In the example you don't have two field structures on the same set. The joint set is $S=BbbRcupinfty$. You have one field structure on the set $Ssetminusinfty$ and another on $Ssetminus0$. They happen to share the multiplicative group, but the zero of one is outside the other field altogether.
– Jyrki Lahtonen
Jul 26 at 20:24
 |Â
show 16 more comments
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3
This is not clear. Can you give a detailed example of the structure you have in mind?
– lulu
Jul 26 at 12:13
4
In particular, how could there be two different additive zeroes? Since $0_itimes x=0_i$ for all $x$ in the field we see that $0_1times 0_2=0_2times 0_1$ is both $0_1$ and $0_2$.
– lulu
Jul 26 at 12:20
4
Not following. As a general rule, it's a bad idea to talk about algebraic structures with no examples in mind. The formalism is meant to generalize the properties you observe in particular examples. Here, I don't think you have any actual examples in mind. If you do, please edit your post to describe one or two examples in detail.
– lulu
Jul 26 at 12:36
2
Again, not following. If you have a precise example, please edit your post to include a detailed discussion of it.
– lulu
Jul 26 at 12:49
1
In the example you don't have two field structures on the same set. The joint set is $S=BbbRcupinfty$. You have one field structure on the set $Ssetminusinfty$ and another on $Ssetminus0$. They happen to share the multiplicative group, but the zero of one is outside the other field altogether.
– Jyrki Lahtonen
Jul 26 at 20:24