1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
|
import numpy as np
def check_associativity(cayley: np.array, semigroup: list) -> bool:
n = cayley.shape[0]
for i in range(n):
for j in range(n):
for k in range(n):
q = semigroup.index(cayley[i, k])
p = semigroup.index(cayley[j, i])
if cayley[p, k] != cayley[j, q]:
return False
return True
def get_right_ideal(element, semigroup, cayley):
i = semigroup.index(element)
return set(cayley[i])
def get_left_ideal(element, semigroup, cayley):
i = semigroup.index(element)
return set(cayley.T[i])
def get_semigroup_ideals(semigroup, cayley):
right_ideals = {}
left_ideals = {}
for element in semigroup:
right_ideals[element] = get_right_ideal(element, semigroup, cayley)
left_ideals[element] = get_left_ideal(element, semigroup, cayley)
return right_ideals, left_ideals
def create_table(semigroup, translations):
cayley = []
for i in semigroup:
row = []
for j in semigroup:
word = i + j
word = translate_word(word, translations)
row.append(word)
cayley.append(row)
return np.array(cayley)
def translate_word(word, translations):
while True:
tmp = word
for key, val in translations:
word = word.replace(key, val)
if tmp == word:
break
return word
def create_grin_relation(semigroup, right_ideals, left_ideals):
n = len(semigroup)
r = np.zeros((n, n), dtype=bool)
l = np.zeros((n, n), dtype=bool)
for i in range(n):
for j in range(n):
element1 = semigroup[i]
element2 = semigroup[j]
r[i][j] = (right_ideals[element1] == right_ideals[element2])
l[i][j] = (left_ideals[element1] == left_ideals[element2])
return r, l
def dfs(graph, used, v, l, is_transposed=False):
used[v] = True
if is_transposed:
l.append(v)
for i in range(len(graph)):
if not(used[i]) and graph[v][i]:
dfs(graph, used, i, l, is_transposed)
if not is_transposed:
l.append(v)
def get_egg_box(grin_relation, semigroup):
n = len(grin_relation)
used = np.zeros(n)
component = []
order = []
for i in range(n):
if not used[i]:
dfs(grin_relation, used, i, order)
used = np.zeros(n)
egg_box = []
for i in range(n):
v = order[n - 1 - i]
if not used[v]:
dfs(grin_relation.T, used, v, component, True)
egg_box.append(component.copy())
component.clear()
return [[semigroup[elem] for elem in egg] for egg in egg_box]
def print_cayley(semigroup, cayley):
n = len(semigroup)
m = max(np.char.str_len(cayley.flatten()))
print(' ' * (m + 2) + '\u2502', end='')
for i in range(n):
print(semigroup[i].rjust(m + 1, ' ') + ' ', end='')
print()
print('\u2500' * (m + 2) + '\u253C' + '\u2500' * ((m + 2) * n))
for i in range(n):
print(semigroup[i].rjust(m + 1, ' ') + ' \u2502', end='')
for j in range(n):
print(cayley[i, j].rjust(m + 1, ' '), end=' ')
print()
def print_ideals(semigroup, right_ideals, left_ideals):
for element in semigroup:
r = right_ideals[element]
l = right_ideals[element]
print(f"({element}] = {{{', '.join(r)}}}")
print(f"[{element}) = {{{', '.join(l)}}}")
print(f"[{element}] = {{{', '.join(r | l)}}}")
def make_semigroup():
print("Введите элементы полугруппы:")
semigroup = set(input().split())
print("Введите количество преобразований:")
k = int(input())
print(f"Введите преобразования:")
translations = []
for _ in range(k):
key, val = input().split()
translations.append((key, val))
while True:
new_elements = set()
for i in semigroup:
for j in semigroup:
word = i + j
word = translate_word(word, translations)
new_elements.add(word)
tmp = semigroup.copy()
semigroup |= new_elements
if tmp == semigroup:
break
semigroup = list(semigroup)
semigroup.sort()
return semigroup, translations
def main():
semigroup, translations = make_semigroup()
print("Полугруппа:")
print(semigroup)
print("Таблица Кэли:")
cayley = create_table(semigroup, translations)
print_cayley(semigroup, cayley)
if not check_associativity(cayley, semigroup):
print("Ассоциативность для данной таблицы Кэли не выполняется")
return
print("Отношение Грина:")
right_ideals, left_ideals = get_semigroup_ideals(semigroup, cayley)
r, l = create_grin_relation(semigroup, right_ideals, left_ideals)
d = r + l
print(d.astype(int))
print("Идеалы:")
print_ideals(semigroup, right_ideals, left_ideals)
print("egg-box диаграмма:")
egg_box = get_egg_box(d, semigroup)
print(egg_box)
if __name__ == "__main__":
main()
|
