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I have a set of variables $x_1,...,x_n$ and $y_1,...,y_m$ for $n,m >=1$.

Now I can build all quadratic monomials of the form $x_i y_j$ and $y_j x_i$ (but we do not have $_xi y_j= y_j x_i$ as we calcualte in the non-commutative polynomial ring). But something like $x_i x_j$ is not allowed as after an $x_i$ there must come and $y_j$ and after an $y_i$ there must come an $x_i$.

Now I want with Sage the list of all possible relations of the form $w_1 \pm w_2 \pm w_3 \cdots$ such that all $w_i$ are different quadratic relations that all start either with a $x_i$ or a $y_j$.

For example for $n=2$ and $m=1$, possible relations are (I hope I did not forget any relation) : $x_1 y_1, x_1 y_1-x_2 y_2, x_1 y_2 + x_2 y_2 , x_2 y_1,y_1 x_1, y_1 x_2, y_1 x_1 - y_1 x_2 , y_1x_1+y_1 x_2$.

I am not sure how to do this in an easy way with Sage, but maybe someone knows a simple trick.

Thanks for any help.

I have a set of variables $x_1,...,x_n$ and $y_1,...,y_m$ for $n,m >=1$.

Now I can build all quadratic monomials of the form $x_i y_j$ and $y_j x_i$ (but we do not have $_xi $x_i y_j= y_j x_i$ as we calcualte in the non-commutative polynomial ring). But something like $x_i x_j$ is not allowed as after an $x_i$ there must come and $y_j$ and after an $y_i$ there must come an $x_i$.

Now I want with Sage the list of all possible relations of the form $w_1 \pm w_2 \pm w_3 \cdots$ such that all $w_i$ are different quadratic relations that all start either with a $x_i$ or a $y_j$.

For example for $n=2$ and $m=1$, possible relations are (I hope I did not forget any relation) : $x_1 y_1, x_1 y_1-x_2 y_2, x_1 y_2 + x_2 y_2 , x_2 y_1,y_1 x_1, y_1 x_2, y_1 x_1 - y_1 x_2 , y_1x_1+y_1 x_2$.

I am not sure how to do this in an easy way with Sage, but maybe someone knows a simple trick.

Thanks for any help.

I have a set of variables $x_1,...,x_n$ and $y_1,...,y_m$ for $n,m >=1$.

Now I can build all quadratic monomials of the form $x_i y_j$ and $y_j x_i$ (but we do not have $x_i y_j= y_j x_i$ as we calcualte in the non-commutative polynomial ring). But something like $x_i x_j$ is not allowed as after an $x_i$ there must come and an $y_j$ and after an $y_i$ there must come an $x_i$.

Now I want with Sage the list of all possible relations of the form $w_1 \pm w_2 \pm w_3 \cdots$ such that all $w_i$ are different quadratic relations that all start either with a $x_i$ or a $y_j$.

For example for $n=2$ and $m=1$, possible relations are (I hope I did not forget any relation) : $x_1 y_1, x_1 y_1-x_2 y_2, x_1 y_2 + x_2 y_2 , x_2 y_1,y_1 x_1, y_1 x_2, y_1 x_1 - y_1 x_2 , y_1x_1+y_1 x_2$.

I am not sure how to do this in an easy way with Sage, but maybe someone knows a simple trick.

Thanks for any help.