Ring-like strusture with non associative addition












0












$begingroup$


There is this structure i found which is the set of continuous maps from [-1, 1]^n into itself, endowed with a "sum" which is the pointwise sum of two functions divided by 2, and a "product" which is the pointwise multiplication of two functions, where the product vector is obtained by component-wise multiplication.



I found that:



ABOUT THE SUM:



sum IS NOT associative



sum IS commutative



a + a = a (sum is idempotent)



for all a there exists a unique b such that a + b = 0



NB: by zero I meant the constant null function, not addition's neutral element (which doesn't exist). This property is meaningful cause it only holds for zero.



ABOUT THE PRODUCT:



prpduct is associative



product is commutative



it's got a unit, the constant (1,1,...,1) function, which is the only invertible element together with the other combinations of 1's and -1's stuffed in a vector
product distributes over addition



Question is: has this got any name? Are there any publications dealing with similar structures? I found some on nonassociative rings (which maybe are called somethig else), i mean ring with non associative multiplication, but I didn't find anything about structures like this.



Post scriptum: there could also be an additional operation: composition.



ABOUT COMPOSITION:



associative



got identity element (x --> x)



right-distributes over both addition and multiplication










share|cite|improve this question











$endgroup$












  • $begingroup$
    You haven't really changed the ring. You've just taken the associative operation and willfully made it not associative. What is the reason for doing this?
    $endgroup$
    – Matt Samuel
    Dec 9 '18 at 20:25










  • $begingroup$
    to keep it C([-1,1]^n, [-1, 1]^n)
    $endgroup$
    – ReLonzo
    Dec 9 '18 at 22:09






  • 1




    $begingroup$
    It's possible you'll get some responses, but there is a better chance if you format your question properly. math.meta.stackexchange.com/questions/5020/…
    $endgroup$
    – Matt Samuel
    Dec 10 '18 at 3:08
















0












$begingroup$


There is this structure i found which is the set of continuous maps from [-1, 1]^n into itself, endowed with a "sum" which is the pointwise sum of two functions divided by 2, and a "product" which is the pointwise multiplication of two functions, where the product vector is obtained by component-wise multiplication.



I found that:



ABOUT THE SUM:



sum IS NOT associative



sum IS commutative



a + a = a (sum is idempotent)



for all a there exists a unique b such that a + b = 0



NB: by zero I meant the constant null function, not addition's neutral element (which doesn't exist). This property is meaningful cause it only holds for zero.



ABOUT THE PRODUCT:



prpduct is associative



product is commutative



it's got a unit, the constant (1,1,...,1) function, which is the only invertible element together with the other combinations of 1's and -1's stuffed in a vector
product distributes over addition



Question is: has this got any name? Are there any publications dealing with similar structures? I found some on nonassociative rings (which maybe are called somethig else), i mean ring with non associative multiplication, but I didn't find anything about structures like this.



Post scriptum: there could also be an additional operation: composition.



ABOUT COMPOSITION:



associative



got identity element (x --> x)



right-distributes over both addition and multiplication










share|cite|improve this question











$endgroup$












  • $begingroup$
    You haven't really changed the ring. You've just taken the associative operation and willfully made it not associative. What is the reason for doing this?
    $endgroup$
    – Matt Samuel
    Dec 9 '18 at 20:25










  • $begingroup$
    to keep it C([-1,1]^n, [-1, 1]^n)
    $endgroup$
    – ReLonzo
    Dec 9 '18 at 22:09






  • 1




    $begingroup$
    It's possible you'll get some responses, but there is a better chance if you format your question properly. math.meta.stackexchange.com/questions/5020/…
    $endgroup$
    – Matt Samuel
    Dec 10 '18 at 3:08














0












0








0





$begingroup$


There is this structure i found which is the set of continuous maps from [-1, 1]^n into itself, endowed with a "sum" which is the pointwise sum of two functions divided by 2, and a "product" which is the pointwise multiplication of two functions, where the product vector is obtained by component-wise multiplication.



I found that:



ABOUT THE SUM:



sum IS NOT associative



sum IS commutative



a + a = a (sum is idempotent)



for all a there exists a unique b such that a + b = 0



NB: by zero I meant the constant null function, not addition's neutral element (which doesn't exist). This property is meaningful cause it only holds for zero.



ABOUT THE PRODUCT:



prpduct is associative



product is commutative



it's got a unit, the constant (1,1,...,1) function, which is the only invertible element together with the other combinations of 1's and -1's stuffed in a vector
product distributes over addition



Question is: has this got any name? Are there any publications dealing with similar structures? I found some on nonassociative rings (which maybe are called somethig else), i mean ring with non associative multiplication, but I didn't find anything about structures like this.



Post scriptum: there could also be an additional operation: composition.



ABOUT COMPOSITION:



associative



got identity element (x --> x)



right-distributes over both addition and multiplication










share|cite|improve this question











$endgroup$




There is this structure i found which is the set of continuous maps from [-1, 1]^n into itself, endowed with a "sum" which is the pointwise sum of two functions divided by 2, and a "product" which is the pointwise multiplication of two functions, where the product vector is obtained by component-wise multiplication.



I found that:



ABOUT THE SUM:



sum IS NOT associative



sum IS commutative



a + a = a (sum is idempotent)



for all a there exists a unique b such that a + b = 0



NB: by zero I meant the constant null function, not addition's neutral element (which doesn't exist). This property is meaningful cause it only holds for zero.



ABOUT THE PRODUCT:



prpduct is associative



product is commutative



it's got a unit, the constant (1,1,...,1) function, which is the only invertible element together with the other combinations of 1's and -1's stuffed in a vector
product distributes over addition



Question is: has this got any name? Are there any publications dealing with similar structures? I found some on nonassociative rings (which maybe are called somethig else), i mean ring with non associative multiplication, but I didn't find anything about structures like this.



Post scriptum: there could also be an additional operation: composition.



ABOUT COMPOSITION:



associative



got identity element (x --> x)



right-distributes over both addition and multiplication







abstract-algebra ring-theory associativity






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 9 '18 at 16:55







ReLonzo

















asked Dec 9 '18 at 16:38









ReLonzoReLonzo

13




13












  • $begingroup$
    You haven't really changed the ring. You've just taken the associative operation and willfully made it not associative. What is the reason for doing this?
    $endgroup$
    – Matt Samuel
    Dec 9 '18 at 20:25










  • $begingroup$
    to keep it C([-1,1]^n, [-1, 1]^n)
    $endgroup$
    – ReLonzo
    Dec 9 '18 at 22:09






  • 1




    $begingroup$
    It's possible you'll get some responses, but there is a better chance if you format your question properly. math.meta.stackexchange.com/questions/5020/…
    $endgroup$
    – Matt Samuel
    Dec 10 '18 at 3:08


















  • $begingroup$
    You haven't really changed the ring. You've just taken the associative operation and willfully made it not associative. What is the reason for doing this?
    $endgroup$
    – Matt Samuel
    Dec 9 '18 at 20:25










  • $begingroup$
    to keep it C([-1,1]^n, [-1, 1]^n)
    $endgroup$
    – ReLonzo
    Dec 9 '18 at 22:09






  • 1




    $begingroup$
    It's possible you'll get some responses, but there is a better chance if you format your question properly. math.meta.stackexchange.com/questions/5020/…
    $endgroup$
    – Matt Samuel
    Dec 10 '18 at 3:08
















$begingroup$
You haven't really changed the ring. You've just taken the associative operation and willfully made it not associative. What is the reason for doing this?
$endgroup$
– Matt Samuel
Dec 9 '18 at 20:25




$begingroup$
You haven't really changed the ring. You've just taken the associative operation and willfully made it not associative. What is the reason for doing this?
$endgroup$
– Matt Samuel
Dec 9 '18 at 20:25












$begingroup$
to keep it C([-1,1]^n, [-1, 1]^n)
$endgroup$
– ReLonzo
Dec 9 '18 at 22:09




$begingroup$
to keep it C([-1,1]^n, [-1, 1]^n)
$endgroup$
– ReLonzo
Dec 9 '18 at 22:09




1




1




$begingroup$
It's possible you'll get some responses, but there is a better chance if you format your question properly. math.meta.stackexchange.com/questions/5020/…
$endgroup$
– Matt Samuel
Dec 10 '18 at 3:08




$begingroup$
It's possible you'll get some responses, but there is a better chance if you format your question properly. math.meta.stackexchange.com/questions/5020/…
$endgroup$
– Matt Samuel
Dec 10 '18 at 3:08










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