Description
Elements of sets may be any immutable object, such as integers, ring elements and lists of such. Ideals may also be elements of sets.
i1 : A = set {1,2};
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i2 : R = QQ[a..d];
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i3 : B = set{a^2-b*c,b*d}
2
o3 = set {a - b*c, b*d}
o3 : Set
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Set operations, such as
membership,
union,
intersection,
difference,
Cartesian product,
Cartesian power, and
subset are available. For example,
i4 : toList B
2
o4 = {a - b*c, b*d}
o4 : List
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i5 : member(1,A)
o5 = true
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i6 : member(-b*c+a^2,B)
o6 = true
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i7 : A ** A
o7 = set {(1, 1), (1, 2), (2, 1), (2, 2)}
o7 : Set
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i8 : A^**2
o8 = set {(1, 1), (1, 2), (2, 1), (2, 2)}
o8 : Set
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i9 : set{1,3,2} - set{1}
o9 = set {2, 3}
o9 : Set
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i10 : set{4,5} + set{5,6}
o10 = set {4, 5, 6}
o10 : Set
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i11 : set{4,5} * set{5,6}
o11 = set {5}
o11 : Set
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i12 : set{1,3,2} === set{1,2,3}
o12 = true
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Ideals in Macaulay2 come equipped with a specific sequence of generators, so the following two ideals are not considered strictly equal, and thus the set containing them will appear to have two elements.
i13 : I = ideal(a,b); J = ideal(b,a);
o13 : Ideal of R
o14 : Ideal of R
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i15 : I == J
o15 = true
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i16 : I === J
o16 = false
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i17 : C = set(ideal(a,b),ideal(b,a))
o17 = set {ideal (a, b), ideal (b, a)}
o17 : Set
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However, if you
trim the ideals, then the generating sets will be the same, and so the set containing them will have one element.
i18 : C1 = set(trim ideal(a,b),trim ideal(b,a))
o18 = set {ideal (b, a)}
o18 : Set
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A set is implemented as a
HashTable, whose keys are the elements of the set, and whose values are all 1. In particular, this means that two objects are considered the same exactly when they are strictly equal, according to
===.