├── .gitignore ├── absalg ├── __init__.py ├── Set.py ├── Function.py └── Group.py ├── setup.py ├── test ├── set_test.py ├── function_test.py └── group_test.py ├── README.md └── COPYING /.gitignore: -------------------------------------------------------------------------------- 1 | *.pyc 2 | build 3 | *.sh 4 | -------------------------------------------------------------------------------- /absalg/__init__.py: -------------------------------------------------------------------------------- 1 | import Set 2 | from Set import * 3 | import Function 4 | from Function import * 5 | import Group 6 | from Group import * 7 | -------------------------------------------------------------------------------- /setup.py: -------------------------------------------------------------------------------- 1 | # Setup script 2 | 3 | from distutils.core import setup 4 | 5 | setup(name='absalg', 6 | version='0.0.1', 7 | description='Abstract Algebra in Python', 8 | author='Naftali Harris', 9 | author_email='naftali@stanford.edu', 10 | packages=['absalg'], 11 | ) 12 | -------------------------------------------------------------------------------- /test/set_test.py: -------------------------------------------------------------------------------- 1 | import unittest 2 | from absalg.Set import * 3 | 4 | class test_set(unittest.TestCase): 5 | def test_basics(self): 6 | for n in range(10): 7 | s = Set(range(n)) 8 | t = Set(xrange(n)) 9 | self.assertEquals(s, s) 10 | self.assertEquals(s, t) 11 | self.assertEquals(t, s) 12 | self.assertEquals(t, t) 13 | self.assertEquals(len(s), n) 14 | 15 | def test_product(self): 16 | for n in range(10): 17 | s = Set(range(n)) 18 | self.assertEquals(s * s, \ 19 | Set((x, y) for x in range(n) for y in range(n))) 20 | 21 | self.assertEquals(Set(range(10)) * Set([]), Set([])) 22 | 23 | if __name__ == "__main__": 24 | unittest.main() 25 | -------------------------------------------------------------------------------- /absalg/Set.py: -------------------------------------------------------------------------------- 1 | """ 2 | Implementation of Set 3 | """ 4 | 5 | class Set(frozenset): 6 | """ 7 | Definition of a Set 8 | 9 | It's important that Set be a subclass of frozenset, (not set), because: 10 | 1) it makes Set immutable 11 | 2) it allows Set to contains Sets 12 | """ 13 | def __mul__(self, other): 14 | """Cartesian product""" 15 | if not isinstance(other, Set): 16 | raise TypeError("One of the objects is not a set") 17 | return Set((x, y) for x in self for y in other) 18 | 19 | def pick(self): 20 | """Return an arbitrary element. (The finite Axiom of Choice is true!)""" 21 | 22 | if len(self) == 0: 23 | raise KeyError("This is an empty set") 24 | 25 | for item in self: break 26 | return item 27 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | This package attempts to implement finite abstract algebra in python. 2 | It is written by Naftali Harris and licensed under the GNU General Public 3 | License, Version 3. 4 | 5 | INSTALLATION 6 | ============ 7 | 8 | $ sudo python setup.py install 9 | 10 | 11 | USAGE 12 | ===== 13 | 14 | $ python 15 | Python 2.7.2+ (default, Oct 4 2011, 20:03:08) 16 | [GCC 4.6.1] on linux2 17 | Type "help", "copyright", "credits" or "license" for more information. 18 | >>> from absalg import * 19 | >>> print Zn(2) * Zn(2) 20 | e: (0, 0) 21 | a: (0, 1) 22 | b: (1, 0) 23 | c: (1, 1) 24 | 25 | | e | a | b | c | 26 | ---+---+---+---+---+ 27 | e | e | a | b | c | 28 | ---+---+---+---+---+ 29 | a | a | e | c | b | 30 | ---+---+---+---+---+ 31 | b | b | c | e | a | 32 | ---+---+---+---+---+ 33 | c | c | b | a | e | 34 | ---+---+---+---+---+ 35 | 36 | >>> 37 | 38 | 39 | LICENSE INFO 40 | ============ 41 | 42 | See "COPYING" 43 | -------------------------------------------------------------------------------- /test/function_test.py: -------------------------------------------------------------------------------- 1 | import unittest 2 | from absalg.Set import Set 3 | from absalg.Function import * 4 | 5 | class test_function(unittest.TestCase): 6 | def test_basics(self): 7 | s = Set([0, 1, 2, 3]) 8 | t = Set([1, 2, 3, 4]) 9 | f = Function(s, t, lambda x: x + 1) 10 | str(f) 11 | for x in range(4): 12 | self.assertEquals(f(x), x + 1) 13 | self.assertEquals(f, f) 14 | 15 | with self.assertRaises(TypeError): 16 | Function([0, 1, 2, 3], t, lambda x: x + 1) 17 | with self.assertRaises(TypeError): 18 | Function(s, [1, 2, 3, 4], lambda x: x + 1) 19 | with self.assertRaises(TypeError): 20 | Function(s, t, lambda x: x + 2) 21 | 22 | def test_simple(self): 23 | s = Set([0, 1, 2, 3]) 24 | t = Set([1, 2, 3, 4]) 25 | u = Set([0, 1, 2, 3, 4, 5]) 26 | f = Function(s, t, lambda x: x + 1) 27 | g = Function(t, u, lambda x: x + 1) 28 | h = g.compose(f) 29 | i = Function(Set([-1, 0, 1]), Set([0, 1]), lambda x: abs(x)) 30 | 31 | self.assertEquals(f.is_surjective(), True) 32 | self.assertEquals(f.is_injective(), True) 33 | self.assertEquals(f.is_bijective(), True) 34 | self.assertEquals(f.image(), t) 35 | str(f) 36 | 37 | self.assertEquals(g.is_surjective(), False) 38 | self.assertEquals(g.is_injective(), True) 39 | self.assertEquals(g.is_bijective(), False) 40 | self.assertEquals(g.image(), Set([2, 3, 4, 5])) 41 | str(g) 42 | 43 | self.assertEquals(h.is_surjective(), False) 44 | self.assertEquals(h.is_injective(), True) 45 | self.assertEquals(h.is_bijective(), False) 46 | self.assertEquals(h.image(), Set([2, 3, 4, 5])) 47 | str(h) 48 | 49 | self.assertEquals(i.is_surjective(), True) 50 | self.assertEquals(i.is_injective(), False) 51 | self.assertEquals(i.is_bijective(), False) 52 | self.assertEquals(i.image(), Set([0, 1])) 53 | str(i) 54 | 55 | with self.assertRaises(ValueError): 56 | i.compose(f) 57 | with self.assertRaises(ValueError): 58 | f.compose(i) 59 | 60 | def test_identity(self): 61 | s = Set(["las", 3, "ksjfdlka"]) 62 | ID = identity(s) 63 | str(ID) 64 | for item in s: 65 | self.assertEquals(ID(item), item) 66 | 67 | if __name__ == "__main__": 68 | unittest.main() 69 | -------------------------------------------------------------------------------- /absalg/Function.py: -------------------------------------------------------------------------------- 1 | """ 2 | Definition of a function 3 | """ 4 | 5 | from Set import Set 6 | 7 | class Function: 8 | """Definition of a finite function""" 9 | def __init__(self, domain, codomain, function): 10 | """ 11 | Initialize the function and check that it is well-formed. 12 | 13 | This method can be overwritten by subclasses of Function, so that for 14 | example GroupHomomorphisms can be between Groups, rather than Sets. 15 | """ 16 | if not isinstance(domain, Set): 17 | raise TypeError("Domain must be a Set") 18 | if not isinstance(codomain, Set): 19 | raise TypeError("Codomain must be a Set") 20 | if not all(function(elem) in codomain for elem in domain): 21 | raise TypeError("Function returns some value outside of codomain") 22 | 23 | self.domain = domain 24 | self.codomain = codomain 25 | self.function = function 26 | 27 | def __call__(self, elem): 28 | if elem not in self.domain: 29 | raise TypeError("Function must be called on elements of the domain") 30 | return self.function(elem) 31 | 32 | def __hash__(self): 33 | """Returns the hash of self""" 34 | 35 | # Need to be a little careful, since self.domain and self.codomain are 36 | # often the same, and we don't want to cancel out their hashes by xoring 37 | # them against each other. 38 | # 39 | # Also, functions we consider equal, like lambda x: x + 1, and 40 | # def jim(x): return x + 1, have different hashes, so we can't include 41 | # the hash of self.function. 42 | # 43 | # Finally, we should make sure that if you switch the domain and 44 | # codomain, the hash will (usually) change, so you can't just add or 45 | # multiply the hashes together. 46 | 47 | return hash(self.domain) + 2 * hash(self.codomain) 48 | 49 | def __eq__(self, other): 50 | if not isinstance(other, Function): 51 | return False 52 | 53 | return id(self) == id(other) or ( \ 54 | self.domain == other.domain and \ 55 | self.codomain == other.codomain and \ 56 | all(self(elem) == other(elem) for elem in self.domain) ) 57 | 58 | def __ne__(self, other): 59 | return not self == other 60 | 61 | def _image(self): 62 | """The literal image of the function""" 63 | return Set(self(elem) for elem in self.domain) 64 | 65 | def image(self): 66 | """ 67 | The API image of the function; can change depending on the subclass. 68 | 69 | For example, GroupHomomorphisms return the image as a Group, not a Set. 70 | """ 71 | return self._image() 72 | 73 | def __str__(self): 74 | """Pretty outputing of functions""" 75 | 76 | # Figure out formatting 77 | maxlen = max(len(str(x)) for x in self.domain) if self.domain else 0 78 | formatstr1 = "{0:<%d} -> {1}\n" % maxlen 79 | formatstr2 = "{0:<%d}{1}\n" % (maxlen + 4) 80 | nothit = self.codomain - self._image() 81 | 82 | return("".join(formatstr1.format(x, self(x)) for x in self.domain) + \ 83 | "".join(formatstr2.format("", y) for y in nothit)) 84 | 85 | def is_surjective(self): 86 | # Need to make self.domain into a Set, since it might not be in 87 | # subclasses of Function 88 | return self._image() == Set(self.codomain) 89 | 90 | def is_injective(self): 91 | return len(self._image()) == len(self.domain) 92 | 93 | def is_bijective(self): 94 | return self.is_surjective() and self.is_injective() 95 | 96 | def compose(self, other): 97 | """Returns x -> self(other(x))""" 98 | if not self.domain == other.codomain: 99 | raise ValueError("codomain of other must match domain of self") 100 | return Function(other.domain, self.codomain, lambda x: self(other(x))) 101 | 102 | def new_domains(self, domain, codomain): 103 | return Function(domain, codomain, self.function) 104 | 105 | def identity(s): 106 | """Returns the identity function on the set s""" 107 | if not isinstance(s, Set): 108 | raise TypeError("s must be a set") 109 | return Function(s, s, lambda x: x) 110 | -------------------------------------------------------------------------------- /test/group_test.py: -------------------------------------------------------------------------------- 1 | import unittest 2 | from math import factorial 3 | from absalg.Group import * 4 | 5 | class test_group(unittest.TestCase): 6 | def test_Zn(self): 7 | for n in range(1, 10): 8 | Z = Zn(n) 9 | str(Z) 10 | self.assertEquals(Z.e, GroupElem(0,Z)) 11 | self.assertEquals(len(Z), n) 12 | self.assertTrue(all(a * b == GroupElem((a.elem + b.elem) % n, Z) for a in Z for b in Z)) 13 | self.assertTrue(all(Z.inverse(a) == GroupElem((n - a.elem) % n, Z) for a in Z)) 14 | self.assertTrue(Z.is_abelian()) 15 | self.assertTrue(Z <= Z) 16 | self.assertTrue(Z.is_normal_subgroup(Z)) 17 | self.assertEquals(len(Z/Z), 1) 18 | if n <= 5: # takes a while 19 | self.assertEquals(len(Z * Z), n * n) 20 | self.assertEquals(Z.generate(Z), Z) 21 | 22 | def test_Sn(self): 23 | for n in range(1, 5): 24 | S = Sn(n) 25 | str(S) 26 | self.assertEquals(S.e, GroupElem(tuple(xrange(n)), S)) 27 | self.assertEquals(len(S), factorial(n)) 28 | self.assertTrue(all(S.inverse(a) == GroupElem( \ 29 | tuple(dict((a.elem[j], j) for j in a.elem)[i] \ 30 | for i in range(n)), S) \ 31 | for a in S)) 32 | if n <= 2: 33 | self.assertTrue(S.is_abelian()) 34 | else: 35 | self.assertFalse(S.is_abelian()) 36 | self.assertTrue(S <= S) 37 | self.assertTrue(S.is_normal_subgroup(S)) 38 | self.assertEquals(len(S/S), 1) 39 | if n <= 3: 40 | self.assertEquals(len(S * S), factorial(n)**2) 41 | self.assertEquals(S.generate(S), S) 42 | 43 | def test_subgroups(self): 44 | G = Zn(9) 45 | sgs = G.subgroups() 46 | self.assertEquals(len(sgs), 3) 47 | for H in sgs: 48 | if H.is_normal_subgroup(G): 49 | self.assertEquals(len(G / H) * len(H), len(G)) 50 | 51 | G = Zn(2) * Zn(2) 52 | sgs = G.subgroups() 53 | self.assertEquals(len(sgs), 5) 54 | for H in sgs: 55 | if H.is_normal_subgroup(G): 56 | self.assertEquals(len(G / H) * len(H), len(G)) 57 | 58 | G = Sn(3) 59 | sgs = G.subgroups() 60 | self.assertEquals(len(G.subgroups()), 6) 61 | for H in sgs: 62 | if H.is_normal_subgroup(G): 63 | self.assertEquals(len(G / H) * len(H), len(G)) 64 | 65 | def test_group_elem(self): 66 | V = Zn(2) * Zn(2) 67 | e, a, b, c = tuple(g for g in V) 68 | self.assertEquals(a + b + c, e) 69 | self.assertEquals(a + b, c) 70 | self.assertEquals(b + c, a) 71 | self.assertEquals(a + c, b) 72 | for g in V: 73 | self.assertTrue(g in V) 74 | self.assertEquals(g, g) 75 | self.assertEquals(e * g, g) 76 | self.assertEquals(g * e, g) 77 | self.assertEquals(e + g, g) 78 | self.assertEquals(g + e, g) 79 | self.assertEquals(g * g, e) 80 | self.assertEquals(g + g, e) 81 | self.assertEquals(g ** -1, g) 82 | self.assertEquals(-g, g) 83 | self.assertEquals(-g, g ** -1) 84 | self.assertEquals(g ** 209325, g) 85 | self.assertEquals(g ** -23234, e) 86 | for n in range(-10, 10): 87 | self.assertEquals(g * n, g ** n) 88 | self.assertEquals(n * g, g ** n) 89 | self.assertEquals(n * g, g * n) 90 | for g in [a, b, c]: 91 | self.assertTrue(e != g) 92 | 93 | G = Sn(3) 94 | for g in G: 95 | with self.assertRaises(TypeError): 96 | g + g 97 | with self.assertRaises(TypeError): 98 | g * 2 99 | with self.assertRaises(TypeError): 100 | 2 * g 101 | with self.assertRaises(TypeError): 102 | -g 103 | for H in G.subgroups(): 104 | for g in G: 105 | self.assertEquals(g ** 5, g * g * g * g * g) 106 | for h in H: 107 | self.assertEquals(h ** 2, h * h) 108 | self.assertEquals(h * g, GroupElem(h.elem, G) * g) 109 | self.assertEquals(g * h, g * GroupElem(h.elem, G)) 110 | 111 | def test_generators(self): 112 | for G in [Zn(1), Zn(2), Zn(5), Zn(8), Sn(1), Sn(2), Sn(3), \ 113 | Dn(1), Dn(2), Dn(3), Dn(4)]: 114 | self.assertEquals(G, G.generate(G.generators())) 115 | self.assertEquals(G, G.generate(g.elem for g in G.generators())) 116 | 117 | def test_find_isomorphism(self): 118 | f = Dn(2).find_isomorphism(Zn(2) * Zn(2)) 119 | self.assertTrue(f is not None) 120 | self.assertTrue(f.is_isomorphism()) 121 | self.assertEquals(f.kernel(), Dn(2).generate([Dn(2).e])) 122 | self.assertEquals(f.image(), Zn(2) * Zn(2)) 123 | 124 | self.assertFalse(Dn(12).is_isomorphic(Sn(4))) 125 | self.assertFalse(Sn(3).is_isomorphic(Zn(6))) 126 | self.assertFalse(Sn(3).is_isomorphic(Zn(4))) 127 | 128 | for G in [Zn(1), Zn(2), Zn(5), Sn(1), Sn(3), Dn(1), Dn(4)]: 129 | self.assertTrue(G.is_isomorphic(G)) 130 | f = G.find_isomorphism(G) 131 | self.assertTrue(f is not None) 132 | self.assertTrue(f.is_isomorphism()) 133 | self.assertEquals(f.kernel(), G.generate([G.e])) 134 | self.assertEquals(f.image(), G) 135 | 136 | self.assertTrue(Zn(1).is_isomorphic(Sn(1))) 137 | self.assertTrue(Zn(2).is_isomorphic(Sn(2))) 138 | self.assertTrue(Zn(2).is_isomorphic(Dn(1))) 139 | 140 | def test_cyclic(self): 141 | for n in range(1, 10): 142 | self.assertTrue(Zn(n).is_cyclic()) 143 | 144 | self.assertTrue(Sn(1).is_cyclic()) 145 | self.assertTrue(Sn(2).is_cyclic()) 146 | self.assertFalse(Sn(3).is_cyclic()) 147 | 148 | self.assertTrue(Dn(1).is_cyclic()) 149 | self.assertFalse(Dn(2).is_cyclic()) 150 | self.assertFalse(Dn(3).is_cyclic()) 151 | 152 | 153 | if __name__ == "__main__": 154 | unittest.main() 155 | -------------------------------------------------------------------------------- /absalg/Group.py: -------------------------------------------------------------------------------- 1 | """Group implementation""" 2 | 3 | import itertools 4 | 5 | from Set import Set 6 | from Function import Function 7 | 8 | class GroupElem: 9 | """ 10 | Group element definition 11 | 12 | This is mainly syntactic sugar, so you can write stuff like g * h 13 | instead of group.bin_op(g, h), or group(g, h). 14 | """ 15 | 16 | def __init__(self, elem, group): 17 | if not isinstance(group, Group): 18 | raise TypeError("group is not a Group") 19 | if not elem in group.Set: 20 | raise TypeError("elem is not an element of group") 21 | self.elem = elem 22 | self.group = group 23 | 24 | def __str__(self): 25 | return str(self.elem) 26 | 27 | def __eq__(self, other): 28 | """ 29 | Two GroupElems are equal if they represent the same element, 30 | regardless of the Groups they belong to 31 | """ 32 | 33 | if not isinstance(other, GroupElem): 34 | raise TypeError("other is not a GroupElem") 35 | return self.elem == other.elem 36 | 37 | def __ne__(self, other): 38 | return not self == other 39 | 40 | def __hash__(self): 41 | return hash(self.elem) 42 | 43 | def __mul__(self, other): 44 | """ 45 | If other is a group element, returns self * other. 46 | If other = n is an int, and self is in an abelian group, returns self**n 47 | """ 48 | if self.group.is_abelian() and isinstance(other, (int, long)): 49 | return self ** other 50 | 51 | if not isinstance(other, GroupElem): 52 | raise TypeError("other must be a GroupElem, or an int " \ 53 | "(if self's group is abelian)") 54 | try: 55 | return GroupElem(self.group.bin_op((self.elem, other.elem)), \ 56 | self.group) 57 | # This can return a TypeError in Funcion.__call__ if self and other 58 | # belong to different Groups. So we see if we can make sense of this 59 | # operation the other way around. 60 | except TypeError: 61 | return other.__rmul__(self) 62 | 63 | def __rmul__(self, other): 64 | """ 65 | If other is a group element, returns other * self. 66 | If other = n is an int, and self is in an abelian group, returns self**n 67 | """ 68 | if self.group.is_abelian() and isinstance(other, (int, long)): 69 | return self ** other 70 | 71 | if not isinstance(other, GroupElem): 72 | raise TypeError("other must be a GroupElem, or an int " \ 73 | "(if self's group is abelian)") 74 | 75 | return GroupElem(self.group.bin_op((other.elem, self.elem)), self.group) 76 | 77 | def __add__(self, other): 78 | """Returns self + other for Abelian groups""" 79 | if self.group.is_abelian(): 80 | return self * other 81 | raise TypeError("not an element of an abelian group") 82 | 83 | def __pow__(self, n, modulo=None): 84 | """ 85 | Returns self**n 86 | 87 | modulo is included as an argument to comply with the API, and ignored 88 | """ 89 | if not isinstance(n, (int, long)): 90 | raise TypeError("n must be an int or a long") 91 | 92 | if n == 0: 93 | return self.group.e 94 | elif n < 0: 95 | return self.group.inverse(self) ** -n 96 | elif n % 2 == 1: 97 | return self * (self ** (n - 1)) 98 | else: 99 | return (self * self) ** (n / 2) 100 | 101 | def __neg__(self): 102 | """Returns self ** -1 if self is in an abelian group""" 103 | if not self.group.is_abelian(): 104 | raise TypeError("self must be in an abelian group") 105 | return self ** (-1) 106 | 107 | def __sub__(self, other): 108 | """Returns self * (other ** -1) if self is in an abelian group""" 109 | if not self.group.is_abelian(): 110 | raise TypeError("self must be in an abelian group") 111 | return self * (other ** -1) 112 | 113 | def order(self): 114 | """Returns the order of self in the Group""" 115 | return len(self.group.generate([self])) 116 | 117 | 118 | class Group: 119 | """Group definition""" 120 | def __init__(self, G, bin_op): 121 | """Create a group, checking group axioms""" 122 | 123 | # Test types 124 | if not isinstance(G, Set): raise TypeError("G must be a set") 125 | if not isinstance(bin_op, Function): 126 | raise TypeError("bin_op must be a function") 127 | if bin_op.codomain != G: 128 | raise TypeError("binary operation must have codomain equal to G") 129 | if bin_op.domain != G * G: 130 | raise TypeError("binary operation must have domain equal to G * G") 131 | 132 | # Test associativity 133 | if not all(bin_op((a, bin_op((b, c)))) == \ 134 | bin_op((bin_op((a, b)), c)) \ 135 | for a, b, c in itertools.product(G, G, G)): 136 | raise ValueError("binary operation is not associative") 137 | 138 | # Find the identity 139 | found_id = False 140 | for e in G: 141 | if all(bin_op((e, a)) == a for a in G): 142 | found_id = True 143 | break 144 | if not found_id: 145 | raise ValueError("G doesn't have an identity") 146 | 147 | # Test for inverses 148 | for a in G: 149 | if not any(bin_op((a, b)) == e for b in G): 150 | raise ValueError("G doesn't have inverses") 151 | 152 | # At this point, we've verified that we have a Group. 153 | # Now determine if the Group is abelian: 154 | self.abelian = all(bin_op((a, b)) == bin_op((b, a)) \ 155 | for a, b in itertools.product(G, G)) 156 | 157 | self.Set = G 158 | self.group_elems = Set(GroupElem(g, self) for g in G) 159 | self.e = GroupElem(e, self) 160 | self.bin_op = bin_op 161 | 162 | def __iter__(self): 163 | """Iterate over the GroupElems in G, returning the identity first""" 164 | yield self.e 165 | for g in self.group_elems: 166 | if g != self.e: yield g 167 | 168 | def __contains__(self, item): 169 | return item in self.group_elems 170 | 171 | def __hash__(self): 172 | return hash(self.Set) ^ hash(self.bin_op) 173 | 174 | def __eq__(self, other): 175 | if not isinstance(other, Group): 176 | return False 177 | 178 | return id(self) == id(other) or \ 179 | (self.Set == other.Set and self.bin_op == other.bin_op) 180 | 181 | def __ne__(self, other): 182 | return not self == other 183 | 184 | def __len__(self): 185 | return len(self.Set) 186 | 187 | def __str__(self): 188 | """Returns the Cayley table""" 189 | 190 | letters = "eabcdfghijklmnopqrstuvwxyz" 191 | if len(self) > len(letters): 192 | return "This group is too big to print a Cayley table" 193 | 194 | # connect letters to elements 195 | toletter = {} 196 | toelem = {} 197 | for letter, elem in zip(letters, self): 198 | toletter[elem] = letter 199 | toelem[letter] = elem 200 | letters = letters[:len(self)] 201 | 202 | # Display the mapping: 203 | result = "\n".join("%s: %s" % (l, toelem[l]) for l in letters) + "\n\n" 204 | 205 | # Make the graph 206 | head = " | " + " | ".join(l for l in letters) + " |" 207 | border = (len(self) + 1) * "---+" + "\n" 208 | result += head + "\n" + border 209 | result += border.join(" %s | " % l + \ 210 | " | ".join(toletter[toelem[l] * toelem[l1]] \ 211 | for l1 in letters) + \ 212 | " |\n" for l in letters) 213 | result += border 214 | return result 215 | 216 | def is_abelian(self): 217 | """Checks if the group is abelian""" 218 | return self.abelian 219 | 220 | def __le__(self, other): 221 | """Checks if self is a subgroup of other""" 222 | if not isinstance(other, Group): 223 | raise TypeError("other must be a Group") 224 | return self.Set <= other.Set and \ 225 | all(self.bin_op((a, b)) == other.bin_op((a, b)) \ 226 | for a, b in itertools.product(self.Set, self.Set)) 227 | 228 | def is_normal_subgroup(self, other): 229 | """Checks if self is a normal subgroup of other""" 230 | return self <= other and \ 231 | all(Set(g * h for h in self) == Set(h * g for h in self) \ 232 | for g in other) 233 | 234 | def __div__(self, other): 235 | """ Returns the quotient group self / other """ 236 | if not other.is_normal_subgroup(self): 237 | raise ValueError("other must be a normal subgroup of self") 238 | G = Set(Set(self.bin_op((g, h)) for h in other.Set) for g in self.Set) 239 | 240 | def multiply_cosets(x): 241 | h = x[0].pick() 242 | return Set(self.bin_op((h, g)) for g in x[1]) 243 | 244 | return Group(G, Function(G * G, G, multiply_cosets)) 245 | 246 | def inverse(self, g): 247 | """Returns the inverse of elem""" 248 | if not g in self.group_elems: 249 | raise TypeError("g isn't a GroupElem in the Group") 250 | for a in self: 251 | if g * a == self.e: 252 | return a 253 | raise RuntimeError("Didn't find an inverse for g") 254 | 255 | def __mul__(self, other): 256 | """Returns the cartesian product of the two groups""" 257 | if not isinstance(other, Group): 258 | raise TypeError("other must be a group") 259 | bin_op = Function((self.Set * other.Set) * (self.Set * other.Set), \ 260 | (self.Set * other.Set), \ 261 | lambda x: (self.bin_op((x[0][0], x[1][0])), \ 262 | other.bin_op((x[0][1], x[1][1])))) 263 | 264 | return Group(self.Set * other.Set, bin_op) 265 | 266 | def generate(self, elems): 267 | """ 268 | Returns the subgroup of self generated by GroupElems elems 269 | 270 | If any of the items aren't already GroupElems, we will try to convert 271 | them to GroupElems before continuing. 272 | 273 | elems must be iterable 274 | """ 275 | 276 | elems = Set(g if isinstance(g, GroupElem) else GroupElem(g, self) \ 277 | for g in elems) 278 | 279 | if not elems <= self.group_elems: 280 | raise ValueError("elems must be a subset of self.group_elems") 281 | if len(elems) == 0: 282 | raise ValueError("elems must have at least one element") 283 | 284 | oldG = elems 285 | while True: 286 | newG = oldG | Set(a * b for a, b in itertools.product(oldG, oldG)) 287 | if oldG == newG: break 288 | else: oldG = newG 289 | oldG = Set(g.elem for g in oldG) 290 | 291 | return Group(oldG, self.bin_op.new_domains(oldG * oldG, oldG)) 292 | 293 | def is_cyclic(self): 294 | """Checks if self is a cyclic Group""" 295 | return any(g.order() == len(self) for g in self) 296 | 297 | def subgroups(self): 298 | """Returns the Set of self's subgroups""" 299 | 300 | old_sgs = Set([self.generate([self.e])]) 301 | while True: 302 | new_sgs = old_sgs | Set(self.generate(list(sg.group_elems) + [g]) \ 303 | for sg in old_sgs for g in self \ 304 | if g not in sg.group_elems) 305 | if new_sgs == old_sgs: break 306 | else: old_sgs = new_sgs 307 | 308 | return old_sgs 309 | 310 | def generators(self): 311 | """ 312 | Returns a list of GroupElems that generate self, with length 313 | at most log_2(len(self)) + 1 314 | """ 315 | 316 | result = [self.e.elem] 317 | H = self.generate(result) 318 | 319 | while len(H) < len(self): 320 | result.append((self.Set - H.Set).pick()) 321 | H = self.generate(result) 322 | 323 | # The identity is always a redundant generator in nontrivial Groups 324 | if len(self) != 1: 325 | result = result[1:] 326 | 327 | return [GroupElem(g, self) for g in result] 328 | 329 | def find_isomorphism(self, other): 330 | """ 331 | Returns an isomorphic GroupHomomorphism between self and other, 332 | or None if self and other are not isomorphic 333 | 334 | Uses Tarjan's algorithm, running in O(n^(log n + O(1))) time, but 335 | runs a lot faster than that if the group has a small generating set. 336 | """ 337 | if not isinstance(other, Group): 338 | raise TypeError("other must be a Group") 339 | 340 | if len(self) != len(other) or self.is_abelian() != other.is_abelian(): 341 | return None 342 | 343 | # Try to match the generators of self with some subset of other 344 | A = self.generators() 345 | for B in itertools.permutations(other, len(A)): 346 | 347 | func = dict(itertools.izip(A, B)) # the mapping 348 | counterexample = False 349 | while not counterexample: 350 | 351 | # Loop through the mapped elements so far, trying to extend the 352 | # mapping or else find a counterexample 353 | noobs = {} 354 | for g, h in itertools.product(func, func): 355 | if g * h in func: 356 | if func[g] * func[h] != func[g * h]: 357 | counterexample = True 358 | break 359 | else: 360 | noobs[g * h] = func[g] * func[h] 361 | 362 | # If we've mapped all the elements of self, then it's a 363 | # homomorphism provided we haven't seen any counterexamples. 364 | if len(func) == len(self): 365 | break 366 | 367 | # Make sure there aren't any collisions before updating 368 | imagelen = len(set(noobs.values()) | set(func.values())) 369 | if imagelen != len(noobs) + len(func): 370 | counterexample = True 371 | func.update(noobs) 372 | 373 | if not counterexample: 374 | return GroupHomomorphism(self, other, lambda x: func[x]) 375 | 376 | return None 377 | 378 | def is_isomorphic(self, other): 379 | """Checks if self and other are isomorphic""" 380 | return bool(self.find_isomorphism(other)) 381 | 382 | 383 | class GroupHomomorphism(Function): 384 | """ 385 | The definition of a Group Homomorphism 386 | 387 | A GroupHomomorphism is a Function between Groups that obeys the group 388 | homomorphism axioms. 389 | """ 390 | 391 | def __init__(self, domain, codomain, function): 392 | """Check types and the homomorphism axioms; records the two groups""" 393 | 394 | if not isinstance(domain, Group): 395 | raise TypeError("domain must be a Group") 396 | if not isinstance(codomain, Group): 397 | raise TypeError("codomain must be a Group") 398 | if not all(function(elem) in codomain for elem in domain): 399 | raise TypeError("Function returns some value outside of codomain") 400 | 401 | if not all(function(a * b) == function(a) * function(b) \ 402 | for a, b in itertools.product(domain, domain)): 403 | raise ValueError("function doesn't satisfy the homomorphism axioms") 404 | 405 | self.domain = domain 406 | self.codomain = codomain 407 | self.function = function 408 | 409 | def kernel(self): 410 | """Returns the kernel of the homomorphism as a Group object""" 411 | G = Set(g.elem for g in self.domain if self(g) == self.codomain.e) 412 | return Group(G, self.domain.bin_op.new_domains(G * G, G)) 413 | 414 | def image(self): 415 | """Returns the image of the homomorphism as a Group object""" 416 | G = Set(g.elem for g in self._image()) 417 | return Group(G, self.codomain.bin_op.new_domains(G * G, G)) 418 | 419 | def is_isomorphism(self): 420 | return self.is_bijective() 421 | 422 | 423 | def Zn(n): 424 | """Returns the cylic group of order n""" 425 | G = Set(range(n)) 426 | bin_op = Function(G * G, G, lambda x: (x[0] + x[1]) % n) 427 | return Group(G, bin_op) 428 | 429 | def Sn(n): 430 | """Returns the symmetric group of order n! """ 431 | G = Set(g for g in itertools.permutations(range(n))) 432 | bin_op = Function(G * G, G, lambda x: tuple(x[0][j] for j in x[1])) 433 | return Group(G, bin_op) 434 | 435 | def Dn(n): 436 | """Returns the dihedral group of order 2n """ 437 | G = Set("%s%d" % (l, x) for l in "RS" for x in xrange(n)) 438 | def multiply_symmetries(x): 439 | l1, l2 = x[0][0], x[1][0] 440 | x1, x2 = int(x[0][1:]), int(x[1][1:]) 441 | if l1 == "R": 442 | if l2 == "R": 443 | return "R%d" % ((x1 + x2) % n) 444 | else: 445 | return "S%d" % ((x1 + x2) % n) 446 | else: 447 | if l2 == "R": 448 | return "S%d" % ((x1 - x2) % n) 449 | else: 450 | return "R%d" % ((x1 - x2) % n) 451 | return Group(G, Function(G * G, G, multiply_symmetries)) 452 | 453 | -------------------------------------------------------------------------------- /COPYING: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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You may not convey a covered 525 | work if you are a party to an arrangement with a third party that is 526 | in the business of distributing software, under which you make payment 527 | to the third party based on the extent of your activity of conveying 528 | the work, and under which the third party grants, to any of the 529 | parties who would receive the covered work from you, a discriminatory 530 | patent license (a) in connection with copies of the covered work 531 | conveyed by you (or copies made from those copies), or (b) primarily 532 | for and in connection with specific products or compilations that 533 | contain the covered work, unless you entered into that arrangement, 534 | or that patent license was granted, prior to 28 March 2007. 535 | 536 | Nothing in this License shall be construed as excluding or limiting 537 | any implied license or other defenses to infringement that may 538 | otherwise be available to you under applicable patent law. 539 | 540 | 12. No Surrender of Others' Freedom. 541 | 542 | If conditions are imposed on you (whether by court order, agreement or 543 | otherwise) that contradict the conditions of this License, they do not 544 | excuse you from the conditions of this License. If you cannot convey a 545 | covered work so as to satisfy simultaneously your obligations under this 546 | License and any other pertinent obligations, then as a consequence you may 547 | not convey it at all. For example, if you agree to terms that obligate you 548 | to collect a royalty for further conveying from those to whom you convey 549 | the Program, the only way you could satisfy both those terms and this 550 | License would be to refrain entirely from conveying the Program. 551 | 552 | 13. Use with the GNU Affero General Public License. 553 | 554 | Notwithstanding any other provision of this License, you have 555 | permission to link or combine any covered work with a work licensed 556 | under version 3 of the GNU Affero General Public License into a single 557 | combined work, and to convey the resulting work. The terms of this 558 | License will continue to apply to the part which is the covered work, 559 | but the special requirements of the GNU Affero General Public License, 560 | section 13, concerning interaction through a network will apply to the 561 | combination as such. 562 | 563 | 14. Revised Versions of this License. 564 | 565 | The Free Software Foundation may publish revised and/or new versions of 566 | the GNU General Public License from time to time. Such new versions will 567 | be similar in spirit to the present version, but may differ in detail to 568 | address new problems or concerns. 569 | 570 | Each version is given a distinguishing version number. If the 571 | Program specifies that a certain numbered version of the GNU General 572 | Public License "or any later version" applies to it, you have the 573 | option of following the terms and conditions either of that numbered 574 | version or of any later version published by the Free Software 575 | Foundation. If the Program does not specify a version number of the 576 | GNU General Public License, you may choose any version ever published 577 | by the Free Software Foundation. 578 | 579 | If the Program specifies that a proxy can decide which future 580 | versions of the GNU General Public License can be used, that proxy's 581 | public statement of acceptance of a version permanently authorizes you 582 | to choose that version for the Program. 583 | 584 | Later license versions may give you additional or different 585 | permissions. However, no additional obligations are imposed on any 586 | author or copyright holder as a result of your choosing to follow a 587 | later version. 588 | 589 | 15. Disclaimer of Warranty. 590 | 591 | THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY 592 | APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT 593 | HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY 594 | OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, 595 | THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 596 | PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM 597 | IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF 598 | ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 599 | 600 | 16. Limitation of Liability. 601 | 602 | IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING 603 | WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS 604 | THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY 605 | GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE 606 | USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF 607 | DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD 608 | PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), 609 | EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF 610 | SUCH DAMAGES. 611 | 612 | 17. Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | --------------------------------------------------------------------------------