The return type depends on the types of x and y. If they have the same type, then usually the return type is the common type of x and y.
Multiplication involving ring elements (including integers, rational numbers, real and complex numbers), ideals, vectors, matrices, modules is generally the usual multiplication, or composition of functions.
The intersection of sets is given by multiplication. See
Set * Set.
i1 : set{hi,you,there} * set{hi,us,here,you}
o1 = set {hi, you}
o1 : Set
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Multiplication involving a list attempts to multiply each element of the list.
i2 : R = QQ[a..d];
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i3 : a * {b,c,d}
o3 = {a*b, a*c, a*d}
o3 : List
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Multiplication of matrices (
Matrix * Matrix) or ring maps is the same as composition.
i4 : f = map(R,R,{b,c,a,d})
o4 = map(R,R,{b, c, a, d})
o4 : RingMap R <--- R
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i5 : g = map(R,R,{(a+b)^2,b^2,c^2,d^2})
2 2 2 2 2
o5 = map(R,R,{a + 2a*b + b , b , c , d })
o5 : RingMap R <--- R
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i6 : f*g
2 2 2 2 2
o6 = map(R,R,{b + 2b*c + c , c , a , d })
o6 : RingMap R <--- R
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i7 : (f*g)(a) == f(g(a))
o7 = true
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Submodules of modules may be produced using multiplication and addition.
i8 : M = R^2; I = ideal(a+b,c);
o9 : Ideal of R
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i10 : N = I*M + a*R^2
o10 = image | a+b 0 c 0 a 0 |
| 0 a+b 0 c 0 a |
2
o10 : R-module, submodule of R
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i11 : isHomogeneous N
o11 = true
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