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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 make_semigroup():
print("Введите элементы полугруппы:")
semigroup = set(input().split())
print("Введите количество преобразований:")
k = int(input())
translations = []
for _ in range(k):
print(f"Введите преобразование: ")
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("egg-box диаграмма:")
egg_box = get_egg_box(d, semigroup)
print(egg_box)
if __name__ == "__main__":
main()
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