# Declare arithmetic with formal variables

I want to create variables in SAGE and and declare arithmetic relations between them. For example, it is easy enough to declare that x1, ..., xn and y1, ..., yn be variables. Is there a way to state that

xi*xj = 0 (in the ring of polynomials, if necessary)

or that

x1 < y1 < x2 < y2 < ...?

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In the algebraic way, you can do something like:

sage: R.<x,y> = PolynomialRing(ZZ) ; R
Multivariate Polynomial Ring in x, y over Integer Ring
sage: I = R.ideal(x*y)
sage: I
Ideal (x*y) of Multivariate Polynomial Ring in x, y over Integer Ring
sage: S = R.quotient_ring(I) ; S
Quotient of Multivariate Polynomial Ring in x, y over Integer Ring by the ideal (x*y)
sage: S(3*x*y+x^2+4*y)
xbar^2 + 4*ybar


In the symbolic way, you can do (but it is much less reliable):

sage: var('x y')
(x, y)
sage: assume(x*y==0)
sage: bool(3*x*y+x^2+4*y == x^2+4*y)
True


But is seems not able to decide simplifications by itself:

sage: (3*x*y+x^2+4*y).full_simplify()
x^2 + (3*x + 4)*y


For the orderings, you can also work symbolically:

sage: var('x y')
(x, y)
sage: assume(x<y)
sage: bool(3*x<3*y)
True

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