def cross_intersect(clan1: 'PP(M x M)', clan2: 'PP(M x M)', _checked=True) -> 'PP(M x M)':
r"""Return the cross-intersection of ``clan1`` and ``clan2``.
The :term:`cross-intersection` of two :term:`clan`\s is a clan that contains the result of
intersecting every :term:`relation` from one clan with every relation from the other clan.
:return: The :term:`binary extension` of :term:`intersection` from the :term:`algebra of
relations` (which inherits it from the :term:`algebra of sets`) to the :term:`algebra of
clans` applied to ``clan1`` and ``clan2``, or `Undef()` if ``clan1`` or ``clan2`` are
not :term:`clan`\s.
"""
if _checked:
if not is_member(clan1):
return _undef.make_or_raise_undef()
if not is_member(clan2):
return _undef.make_or_raise_undef()
else:
assert is_member(clan1)
assert is_member(clan2)
result = _extension.binary_extend(clan1, clan2, _functools.partial(
_sets.intersect, _checked=False), _checked=False)
if not result.is_empty:
result.cache_clan(_mo.CacheStatus.IS)
if clan1.cached_is_functional or clan2.cached_is_functional:
result.cache_functional(_mo.CacheStatus.IS)
if clan1.cached_is_right_functional or clan2.cached_is_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS)
return result
def binary_multi_extend(multiset1: 'P( M x N )', multiset2: 'P( M x N )', op,
_checked=True) -> 'P( M x N )':
r"""Return the :term:`binary extension` of ``op`` from one :term:`algebra` to another algebra.
For this extension, the elements of the extended algebra must be :term:`multiset`\s of the
elements of the original algebra.
:param multiset1: A :term:`multiset` with elements on which ``op`` operates.
:param multiset2: A multiset with elements on which ``op`` operates.
:param op: A :term:`binary operation` that operates on the elements of ``multiset1`` and
``multiset2``.
:return: A multiset that consists of the defined results of ``op`` when executed on all
combinations of the elements of ``multiset1`` and ``multiset2``, or `Undef()` if either
set is not a :class:`~.Multiset`.
"""
if _checked:
if not isinstance(multiset1, _mo.Multiset):
return _ud.make_or_raise_undef()
if not isinstance(multiset2, _mo.Multiset):
return _ud.make_or_raise_undef()
else:
assert multiset1.is_multiset
assert multiset2.is_multiset
def _get_values(_set1, _set2):
return_count = _collections.Counter()
for elem1, multi1 in _set1.data.items():
for elem2, multi2 in _set2.data.items():
result = op(elem1, elem2)
if result is not _ud.Undef():
return_count[result] += multi1 * multi2
return return_count
return _mo.Multiset(_get_values(multiset1, multiset2), direct_load=True)
def big_intersect(set_: 'PP( M )', _checked=True) -> 'P( M )':
"""Return the intersection of all members of ``set_``.
:return: The :term:`intersection` of all members of ``set_`` or `Undef()` if ``set_`` or any of
its members are not instances of :class:`~.Set`.
Example code:
.. code::
from algebraixlib.mathobjects import Set
from algebraixlib.algebras.sets import big_intersect
big_intersect(Set(Set('a', 'b'), Set('b', 'c')))
# Output: Set(Atom('b'))
big_intersect(Set(Set('a', 'b'), 'a'))
# Output: <algebraixlib.undef.Undef at 0x4004978>
"""
if _checked:
if not isinstance(set_, _mo.Set):
return _undef.make_or_raise_undef()
for element in set_:
if not is_member(element):
return _undef.make_or_raise_undef()
else:
assert set_.is_set
return chain_binary_operation(set_, _functools.partial(intersect, _checked=False), is_member)
def get_left(rel: 'P(M x M)', right: '( M )', _checked=True) -> '( M )':
r"""Return the left component of the couplet that has a right component of ``right``.
In general, use with :term:`right-functional` :term:`relation`\s; that is, relations
where all :term:`right component`\s appear at most once.
:return: The :term:`left component` of the :term:`couplet` that has a :term:`right component`
of ``right``, or `Undef()` if there is not exactly one couplet with the right component
``right`` in ``rel`` or ``rel`` is not a :term:`relation`.
"""
if _checked:
if not is_member(rel):
return _undef.make_or_raise_undef2(rel)
if right is _undef.Undef():
return _undef.make_or_raise_undef(2)
right = _mo.auto_convert(right)
else:
assert is_member_or_undef(rel)
assert _mo.is_mathobject_or_undef(right)
if right is _undef.Undef() or rel is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = None
for elem in rel:
assert elem.is_couplet
if elem.right == right:
if result is not None:
return _undef.make_or_raise_undef() # Early Undef() exit if more than one found.
result = elem.left
if result is None:
return _undef.make_or_raise_undef() # Undef() exit if none found.
return result
def compose(multiclan1: 'P(P(M x M) x N)', multiclan2: 'P(P(M x M) x N)',
_checked=True) -> 'P(P(M x M) x N)':
r"""Return the composition of ``multiclan1`` with ``multiclan2``.
:return: The :term:`binary multi-extension` of :term:`composition` from the
:term:`algebra of relations` to the :term:`algebra of multiclans`, applied to
``multiclan1`` and ``multiclan2``, or `Undef()` if ``multiclan1`` or ``multiclan2``
are not :term:`multiclan`\s.
"""
if _checked:
if not is_member(multiclan1):
return _undef.make_or_raise_undef()
if not is_member(multiclan2):
return _undef.make_or_raise_undef()
else:
assert is_member(multiclan1)
assert is_member(multiclan2)
result = _extension.binary_multi_extend(multiclan1, multiclan2, _functools.partial(
_relations.compose, _checked=False), _checked=False)
if not result.is_empty:
result.cache_multiclan(_mo.CacheStatus.IS)
if multiclan1.cached_is_absolute and multiclan2.cached_is_absolute:
result.cache_absolute(_mo.CacheStatus.IS)
if multiclan1.cached_is_functional and multiclan2.cached_is_functional:
result.cache_functional(_mo.CacheStatus.IS)
if multiclan1.cached_is_right_functional and multiclan2.cached_is_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS)
return result
def right_functional_union(rel1: 'P(M x M)',
rel2: 'P(M x M)',
_checked=True) -> 'P(M x M)':
r"""Return the union of ``rel1`` and ``rel2`` if it is right-functional, otherwise
`Undef()`.
:return: The :term:`right-functional union` of the :term:`relation`\s ``rel1`` and
``rel2``; that is, the :term:`union` if the result is :term:`right-functional`,
otherwise `Undef()`. Also return `Undef()` if ``rel1`` or ``rel2`` are not relations.
"""
if _checked:
if not is_member(rel1):
return _undef.make_or_raise_undef2(rel1)
if not is_member(rel2):
return _undef.make_or_raise_undef2(rel2)
else:
assert is_member_or_undef(rel1)
assert is_member_or_undef(rel2)
if rel1 is _undef.Undef() or rel2 is _undef.Undef():
return _undef.make_or_raise_undef(2)
rel_union = _sets.union(rel1, rel2, _checked=False).cache_relation(
_mo.CacheStatus.IS)
if not is_right_functional(rel_union, _checked=False):
return _undef.make_or_raise_undef(2)
return rel_union
def fill_lefts(rel: 'P(M x M)', renames: 'P(M x M)', _checked=True) -> 'P(M x M)':
r"""Return the left components in ``rel`` that are missing in ``renames`` as a diagonal
unioned with ``renames``.
The purpose is to create a :term:`relation` that can be used with the :term:`composition`
operation to change (rename) one or more :term:`left component`\s and leave the rest alone.
:param rel: The :term:`relation` that provides the full :term:`left set`.
:param renames: A relation where the :term:`right component`\s are meant to be
:term:`composition` 'origins' and the :term:`left component`\s composition 'targets'.
:return: A relation that contains all members of ``renames`` unioned with a :term:`diagonal`
that consists of all left components in ``rel`` that are missing in ``renames``.
"""
if _checked:
if not is_member(rel):
return _undef.make_or_raise_undef()
if not is_member(renames):
return _undef.make_or_raise_undef()
else:
assert is_member(rel)
assert is_member(renames)
missing_lefts = _sets.minus(get_lefts(rel, _checked=False),
get_rights(renames, _checked=False), _checked=False)
diag_missing_lefts = diag(*missing_lefts, _checked=False)
result = _sets.union(renames, diag_missing_lefts, _checked=False)
assert result.cached_is_relation
return result
def compose(rel1: 'P(M x M)', rel2: 'P(M x M)', _checked=True) -> 'P(M x M)':
r"""Return the composition of ``rel1`` with ``rel2``.
:return: The :term:`binary extension` of :term:`composition` from the :term:`algebra of
couplets` to the :term:`algebra of relations`, applied to the :term:`relation`\s
``rel1`` and ``rel2``, or `Undef()` if ``rel1`` or ``rel2`` are not relations.
"""
if _checked:
if not is_member(rel1):
return _undef.make_or_raise_undef()
if not is_member(rel2):
return _undef.make_or_raise_undef()
else:
assert is_member(rel1)
assert is_member(rel2)
result = _extension.binary_extend(rel1, rel2, _functools.partial(
_couplets.compose, _checked=False), _checked=False)
if not result.is_empty:
result.cache_relation(_mo.CacheStatus.IS)
if rel1.cached_is_absolute and rel2.cached_is_absolute:
result.cache_absolute(_mo.CacheStatus.IS)
if rel1.cached_is_functional and rel2.cached_is_functional:
result.cache_functional(_mo.CacheStatus.IS)
if rel1.cached_is_right_functional and rel2.cached_is_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS)
return result
def cross_right_functional_union(clan1: 'PP(M x M)', clan2: 'PP(M x M)',
_checked=True) -> 'PP(M x M)':
r"""Return the cross-right-functional union of ``clan1`` and ``clan2``.
The :term:`cross-right-functional union` of two :term:`clan`\s is the :term:`cross-union`
of these clans, but removing all resulting :term:`relation`\s that are not
:term:`right-functional`.
:return: The :term:`binary extension` of the :term:`right-functional union` from the
:term:`algebra of relations` to the :term:`algebra of clans`, applied to ``clan1`` and
``clan2``, or `Undef()` if ``clan1`` or ``clan2`` are not :term:`clan`\s.
"""
if _checked:
if not is_member(clan1):
return _undef.make_or_raise_undef()
if not is_member(clan2):
return _undef.make_or_raise_undef()
else:
assert is_member(clan1)
assert is_member(clan2)
result = _extension.binary_extend(clan1, clan2, _functools.partial(
_relations.right_functional_union, _checked=False), _checked=False)
if not result.is_empty:
result.cache_clan(_mo.CacheStatus.IS)
result.cache_right_functional(_mo.CacheStatus.IS)
if clan1.cached_is_not_functional or clan2.cached_is_not_functional:
result.cache_functional(_mo.CacheStatus.IS_NOT)
return result
def lhs_cross_functional_union(lhs: 'PP( MxM )', rhs: 'PP( MxM )',
_checked=True) -> 'PP(M x M)':
"""Return the :term:`lhs-cross-functional union` ('left join') of ``lhs`` and ``rhs``.
This operation results in a :term:`clan` that contains every :term:`relation` of a
:term:`cross-functional union`, but also contains all relations in ``lhs`` that
are not already part of one of the resulting relations.
:param lhs: All relations in this clan are guaranteed to be represented in the result.
:return: The resulting clan or `Undef()` if ``lhs`` or ``rhs`` are not clans.
"""
if _checked:
if not is_member(lhs):
return _undef.make_or_raise_undef()
if not is_member(rhs):
return _undef.make_or_raise_undef()
else:
assert is_member(lhs)
assert is_member(rhs)
cfu = cross_functional_union(lhs, rhs, _checked=False)
lhs_rest = _mo.Set(
lhs_elem for lhs_elem in lhs
if cross_functional_union(_mo.Set(lhs_elem, direct_load=True), rhs).is_empty)
result = _sets.union(cfu, lhs_rest, _checked=False)
if not result.is_empty:
result.cache_clan(_mo.CacheStatus.IS)
if lhs.cached_is_functional:
result.cache_functional(_mo.CacheStatus.IS)
if lhs.cached_is_not_right_functional or rhs.cached_is_not_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS_NOT)
if not rhs.is_empty and not lhs_rest.is_empty:
result.cache_regular(_mo.CacheStatus.IS_NOT)
return result
def superstrict(multiset1: 'P( M x N )', multiset2: 'P( M X N )',
_checked=True) -> 'P( M X N )':
"""Return ``multiset1`` if ``multiset1`` is a superset of ``multiset2`` or `Undef()` if not.
:return: The :term:`superstriction` of ``multiset1`` and ``multiset2`` (may return
`Undef()`). Also return `Undef()` if ``multiset1`` or ``multiset2`` are not instances
of :class:`~.Multiset`.
"""
if _checked:
if not is_member(multiset1):
return _undef.make_or_raise_undef2(multiset1)
if not is_member(multiset2):
return _undef.make_or_raise_undef2(multiset2)
else:
assert is_member_or_undef(multiset1)
assert is_member_or_undef(multiset2)
if multiset1 is _undef.Undef() or multiset2 is _undef.Undef():
return _undef.make_or_raise_undef(2)
if not is_superset_of(multiset1, multiset2, _checked=False):
return _undef.make_or_raise_undef(2)
if not multiset1.is_empty:
# Multiclan flags:
if multiset1.cached_is_clan:
if multiset2.cached_is_not_absolute:
multiset1.cache_absolute(_mo.CacheStatus.IS_NOT)
if multiset2.cached_is_not_functional:
multiset1.cache_functional(_mo.CacheStatus.IS_NOT)
if multiset2.cached_is_not_right_functional:
multiset1.cache_right_functional(_mo.CacheStatus.IS_NOT)
if multiset2.cached_is_not_regular:
multiset1.cache_regular(_mo.CacheStatus.IS_NOT)
if multiset2.cached_is_not_right_regular:
multiset1.cache_right_regular(_mo.CacheStatus.IS_NOT)
return multiset1
def export_csv(absolute_clan: 'PP(A x A)', file_or_path, column_order=None, sort_key=None):
"""This function takes an absolute, left regular clan and produces a CSV file, where the
left components are in the header line and the rows are the individual relations.
:param absolute_clan: A :class:`algebraixlib.mathobjects.set.Set`. It must be an absolute and
left-regular clan.
:param file_or_path: Either a file path (in this case the CSV data is written to a file at this
location) or a file object (in this case the CSV data is written to its ``.write()``
function).
:param column_order: The optional left set to be exported in the specified order.
:param sort_key: The optional key to be provided to sorted (sorted not called if None).
:return: ``True`` if the CSV export succeeded.
"""
if not _clans.is_absolute_member(absolute_clan) \
and not _multiclans.is_absolute_member(absolute_clan):
return _ud.make_or_raise_undef()
if column_order is None and not absolute_clan.is_left_regular():
return _ud.make_or_raise_undef()
if column_order is None:
# Since this clan is left regular, get first relation to acquire left set.
rel = next(iter(absolute_clan))
# left_set is sorted to guarantee consistent iterations
column_order = sorted([left.value for left in rel.get_left_set()])
# Build list of dictionaries that associates left components with their right components for each relation.
clan_as_list_of_dicts = _convert_clan_to_list_of_dicts(column_order,
(absolute_clan if sort_key is None else
sorted(absolute_clan, key=sort_key)))
# Write that dictionary.
_csv_dict_writer(file_or_path, column_order, clan_as_list_of_dicts)
return True
def superstrict(multiset1: 'P( M x N )',
multiset2: 'P( M X N )',
_checked=True) -> 'P( M X N )':
"""Return ``multiset1`` if ``multiset1`` is a superset of ``multiset2`` or `Undef()` if not.
:return: The :term:`superstriction` of ``multiset1`` and ``multiset2`` (may return
`Undef()`). Also return `Undef()` if ``multiset1`` or ``multiset2`` are not instances
of :class:`~.Multiset`.
"""
if _checked:
if not is_member(multiset1):
return _undef.make_or_raise_undef2(multiset1)
if not is_member(multiset2):
return _undef.make_or_raise_undef2(multiset2)
else:
assert is_member_or_undef(multiset1)
assert is_member_or_undef(multiset2)
if multiset1 is _undef.Undef() or multiset2 is _undef.Undef():
return _undef.make_or_raise_undef(2)
if not is_superset_of(multiset1, multiset2, _checked=False):
return _undef.make_or_raise_undef(2)
if not multiset1.is_empty:
# Multiclan flags:
if multiset1.cached_is_clan:
if multiset2.cached_is_not_absolute:
multiset1.cache_absolute(CacheStatus.IS_NOT)
if multiset2.cached_is_not_functional:
multiset1.cache_functional(CacheStatus.IS_NOT)
if multiset2.cached_is_not_right_functional:
multiset1.cache_right_functional(CacheStatus.IS_NOT)
if multiset2.cached_is_not_regular:
multiset1.cache_regular(CacheStatus.IS_NOT)
if multiset2.cached_is_not_right_regular:
multiset1.cache_right_regular(CacheStatus.IS_NOT)
return multiset1
def chain_binary_operation(set_, binary_op, is_algebra_member):
r"""Chain all elements of ``set_`` with the binary operation ``binary_op`` and return the
result.
:param set_: A :term:`set` of sets or :term:`multiset`\s.
:param binary_op: The operation through which the members of ``set_`` are chained. It must be
commutative and associative.
:param is_algebra_member: The ``is_member()`` function of the :term:`algebra` of which the
elements of ``set_`` must be members.
:return: A member of ``algebra`` that is the result of chaining all elements of ``set_`` with
the :term:`binary operation` ``binary_op``.
"""
if not is_member(set_):
return _undef.make_or_raise_undef2(set_)
if set_.is_empty:
return set_
set_itr = iter(set_)
element1 = next(set_itr)
if not is_algebra_member(element1):
return _undef.make_or_raise_undef()
result = element1
for element in set_itr:
if not is_algebra_member(element):
return _undef.make_or_raise_undef()
result = binary_op(result, element)
return result
def chain_binary_operation(set_, binary_op, is_algebra_member):
r"""Chain all elements of ``set_`` with the binary operation ``binary_op`` and return the
result.
:param set_: A :term:`set` of sets or :term:`multiset`\s.
:param binary_op: The operation through which the members of ``set_`` are chained. It must be
commutative and associative.
:param is_algebra_member: The ``is_member()`` function of the :term:`algebra` of which the
elements of ``set_`` must be members.
:return: A member of ``algebra`` that is the result of chaining all elements of ``set_`` with
the :term:`binary operation` ``binary_op``.
"""
if not is_member(set_):
return _undef.make_or_raise_undef()
if set_.is_empty:
return set_
set_itr = iter(set_)
element1 = next(set_itr)
if not is_algebra_member(element1):
return _undef.make_or_raise_undef()
result = element1
for element in set_itr:
if not is_algebra_member(element):
return _undef.make_or_raise_undef()
result = binary_op(result, element)
return result
def binary_extend(set1: 'P( M )', set2: 'P( M )', op, _checked=True) -> 'P( M )':
r"""Return the :term:`binary extension` of ``op`` from one :term:`algebra` to another algebra.
For this extension, the elements of the extended algebra must be :term:`set`\s of the
elements of the original algebra.
:param set1: A :term:`set` with elements on which ``op`` operates.
:param set2: A set with elements on which ``op`` operates.
:param op: A :term:`binary operation` that operates on the elements of ``set1`` and ``set2``.
:return: A set that consists of the defined results of ``op`` when executed on all combinations
of the elements of ``set1`` and ``set2``, or `Undef()` if either set is not a
:class:`~.Set`.
"""
if _checked:
if not isinstance(set1, _mo.Set):
return _ud.make_or_raise_undef()
if not isinstance(set2, _mo.Set):
return _ud.make_or_raise_undef()
else:
assert set1.is_set
assert set2.is_set
def _get_values(_set1, _set2):
for e1 in _set1:
for e2 in _set2:
result = op(e1, e2)
if result is not _ud.Undef():
yield result
return _mo.Set(_get_values(set1, set2), direct_load=True)
def _to_listgen_check_args(mclan: 'P(P(M x M) x N)', offset: '( A )', limit: '( A )',
_checked: bool=True) -> ():
"""Check the arguments of `multiclan_to_listgen` and `order_slice_to_listgen`. Return a tuple
with the clean values of ``offset`` and ``limit`` or `Undef()` if there was a problem.
"""
if _checked:
if not is_member(mclan):
return _undef.make_or_raise_undef2(mclan)
if isinstance(offset, _mo.Atom) and isinstance(offset.value, int):
pass
elif isinstance(offset, int):
offset = _mo.Atom(limit)
elif offset is _undef.Undef():
return _undef.make_or_raise_undef(2)
else:
return _undef.make_or_raise_undef()
if isinstance(limit, _mo.Atom) \
and (isinstance(limit.value, int) or limit.value == float('inf')):
pass
elif isinstance(limit, int) or limit == float('inf'):
limit = _mo.Atom(limit)
elif limit is _undef.Undef():
return _undef.make_or_raise_undef(2)
else:
return _undef.make_or_raise_undef()
else:
if mclan is _undef.Undef() or offset is _undef.Undef() or limit is _undef.Undef():
return _undef.make_or_raise_undef(2)
assert is_member(mclan)
assert offset.is_atom and isinstance(offset.value, int)
assert limit.is_atom and (isinstance(limit.value, int) or limit.value == float('inf'))
return offset, limit
def get_left(rel: 'P(M x M)', right: '( M )', _checked=True) -> '( M )':
r"""Return the left component of the couplet that has a right component of ``right``.
In general, use with :term:`right-functional` :term:`relation`\s; that is, relations
where all :term:`right component`\s appear at most once.
:return: The :term:`left component` of the :term:`couplet` that has a :term:`right component`
of ``right``, or `Undef()` if there is not exactly one couplet with the right component
``right`` in ``rel`` or ``rel`` is not a :term:`relation`.
"""
if _checked:
if not is_member(rel):
return _undef.make_or_raise_undef2(rel)
if right is _undef.Undef():
return _undef.make_or_raise_undef(2)
right = _mo.auto_convert(right)
else:
assert is_member_or_undef(rel)
assert _mo.is_mathobject_or_undef(right)
if right is _undef.Undef() or rel is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = None
for elem in rel:
assert elem.is_couplet
if elem.right == right:
if result is not None:
return _undef.make_or_raise_undef(
) # Early Undef() exit if more than one found.
result = elem.left
if result is None:
return _undef.make_or_raise_undef() # Undef() exit if none found.
return result
def intersect(multiset1: 'P( M x N )', multiset2: 'P( M x N )', _checked=True) -> 'P( M x N )':
"""Return the multiset intersection of ``multiset1`` with ``multiset2``.
:return: The :term:`multiset intersection` of ``multiset1`` and ``multiset2`` or `Undef()`
if ``multiset1`` or ``multiset2`` are not instances of :class:`~.Multiset`.
"""
if _checked:
if not is_member(multiset1):
return _undef.make_or_raise_undef()
if not is_member(multiset2):
return _undef.make_or_raise_undef()
else:
assert is_member(multiset1)
assert is_member(multiset2)
values = multiset1.data & multiset2.data
result = _mo.Multiset(values)
if not result.is_empty:
# Multiclan flags:
if multiset1.cached_is_multiclan or multiset2.cached_is_multiclan:
result.cache_multiclan(_mo.CacheStatus.IS)
if multiset1.cached_is_absolute or multiset2.cached_is_absolute:
result.cache_absolute(_mo.CacheStatus.IS)
if multiset1.cached_is_functional or multiset2.cached_is_functional:
result.cache_functional(_mo.CacheStatus.IS)
if multiset1.cached_is_right_functional or multiset2.cached_is_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS)
if multiset1.cached_is_regular or multiset2.cached_is_regular:
result.cache_regular(_mo.CacheStatus.IS)
if multiset1.cached_is_right_regular or multiset2.cached_is_right_regular:
result.cache_right_regular(_mo.CacheStatus.IS)
return result
def cross_functional_union(mclan1: 'P(P(M x M) x N)', mclan2: 'P(P(M x M) x N)',
_checked=True) -> 'P(P(M x M) x N)':
r"""Return the :term:`cross-functional union` of ``mclan1`` and ``mclan2``.
:return: The :term:`binary multi-extension` of the :term:`functional union` from the
:term:`algebra of relations` to the :term:`algebra of multiclans`, applied to
``mclan1`` and ``mclan2``, or `Undef()` if ``mclan1`` or ``mclan2`` are
not :term:`multiclan`\s.
"""
if _checked:
if not is_member(mclan1):
return _undef.make_or_raise_undef()
if not is_member(mclan2):
return _undef.make_or_raise_undef()
else:
assert is_member(mclan1)
assert is_member(mclan2)
result = _extension.binary_multi_extend(mclan1, mclan2, _functools.partial(
_relations.functional_union, _checked=False), _checked=False)
if not result.is_empty:
result.cache_multiclan(_mo.CacheStatus.IS)
result.cache_functional(_mo.CacheStatus.IS)
if mclan1.cached_is_not_right_functional or mclan2.cached_is_not_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS_NOT)
return result
def cross_intersect(multiclan1: 'P(P(M x M) x N)', multiclan2: 'PP(M x M)',
_checked=True) -> 'PP(M x M)':
r"""Return the :term:`cross-intersection` of ``multiclan1`` and ``multiclan2``.
:return: The :term:`binary multi-extension` of :term:`intersection` from the :term:`algebra
of relations` (which inherits it from the :term:`algebra of sets`) to the :term:`algebra
of multiclans` applied to ``multiclan1`` and ``multiclan2``, or `Undef()` if
``multiclan1`` or ``multiclan2`` are not :term:`multiclan`\s.
"""
if _checked:
if not is_member(multiclan1):
return _undef.make_or_raise_undef()
if not is_member(multiclan2):
return _undef.make_or_raise_undef()
else:
assert is_member(multiclan1)
assert is_member(multiclan2)
result = _extension.binary_multi_extend(multiclan1, multiclan2, _functools.partial(
_sets.intersect, _checked=False), _checked=False)
if not result.is_empty:
result.cache_multiclan(_mo.CacheStatus.IS)
if multiclan1.cached_is_functional or multiclan2.cached_is_functional:
result.cache_functional(_mo.CacheStatus.IS)
if multiclan1.cached_is_right_functional or multiclan2.cached_is_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS)
return result
def defined_at(mclan: 'P(P(M x M) x N)', left: '( M )', _checked=True):
r"""Return the :term:`relation`\s of ``mclan`` that are defined for ``left``."""
if not is_member(mclan):
return _undef.make_or_raise_undef(2)
if left is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = _extension.unary_multi_extend(mclan,
_functools.partial(
_relations.defined_at,
left=left,
_checked=_checked),
_checked=_checked)
if result is _undef.Undef() or not result:
return _undef.make_or_raise_undef2(result)
result.cache_multiclan(_mo.CacheStatus.IS)
if not result.is_empty:
if mclan.cached_is_functional:
result.cache_functional(_mo.CacheStatus.IS)
if mclan.cached_is_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS)
if mclan.cached_is_regular:
result.cache_regular(_mo.CacheStatus.IS)
if mclan.cached_is_right_regular:
result.cache_right_regular(_mo.CacheStatus.IS)
return result
def big_intersect(set_: 'PP( M )', _checked=True) -> 'P( M )':
"""Return the intersection of all members of ``set_``.
:return: The :term:`intersection` of all members of ``set_`` or `Undef()` if ``set_`` or any of
its members are not instances of :class:`~.Set`.
Example code:
.. code::
from algebraixlib.mathobjects import Set
from algebraixlib.algebras.sets import big_intersect
big_intersect(Set(Set('a', 'b'), Set('b', 'c')))
# Output: Set(Atom('b'))
big_intersect(Set(Set('a', 'b'), 'a'))
# Output: <algebraixlib.undef.Undef at 0x4004978>
"""
if _checked:
if not is_member(set_):
return _undef.make_or_raise_undef2(set_)
for element in set_:
if not is_member(element):
return _undef.make_or_raise_undef(2)
else:
assert is_member_or_undef(set_)
if set_ is _undef.Undef():
return _undef.make_or_raise_undef(2)
return chain_binary_operation(
set_, _functools.partial(intersect, _checked=False), is_member)
def single(set_: _mo.Set):
"""Return the single element of ``set_``.
:return: Return the single element of ``set_``, or `Undef()` if ``set_`` has not exactly one
element or is not a :term:`set` (that is, an instance of :class:`~.Set`).
"""
if not is_member(set_):
return _undef.make_or_raise_undef()
if set_.cardinality == 1:
return next(iter(set_))
return _undef.make_or_raise_undef(2)
def test_make_or_raise_undef(self):
"""Test make_or_raise_undef() together with RaiseOnUndef."""
self.assertEqual(RaiseOnUndef.get_level(), 0)
self.assertIs(make_or_raise_undef(), Undef())
RaiseOnUndef.set_level(1)
self.assertRaises(UndefException, lambda: make_or_raise_undef())
self.assertIs(make_or_raise_undef(2), Undef())
RaiseOnUndef.set_level(2)
self.assertRaises(UndefException, lambda: make_or_raise_undef(2))
RaiseOnUndef.reset()
self.assertIs(make_or_raise_undef(2), Undef())
def union(set1: 'P( M )', set2: 'P( M )', _checked=True) -> 'P( M )':
r"""Return the union of ``set1`` with ``set2``.
:return: The :term:`binary union` of ``set1`` and ``set2`` or `Undef()` if ``set1`` or
``set2`` are not :term:`set`\s (that is, instances of :class:`~.Set`).
"""
# pylint: disable=too-many-branches
if _checked:
if not is_member(set1):
return _undef.make_or_raise_undef()
if not is_member(set2):
return _undef.make_or_raise_undef()
else:
assert is_member(set1)
assert is_member(set2)
values = set1.data.union(set2.data)
result = _mo.Set(values, direct_load=True)
if not result.is_empty:
# Relation flags:
if set1.cached_is_relation and set2.cached_is_relation:
result.cache_relation(_mo.CacheStatus.IS)
if set1.cached_is_absolute and set2.cached_is_absolute:
result.cache_absolute(_mo.CacheStatus.IS)
elif set1.cached_is_not_absolute or set2.cached_is_not_absolute:
result.cache_absolute(_mo.CacheStatus.IS_NOT)
if set1.cached_is_not_functional or set2.cached_is_not_functional:
result.cache_functional(_mo.CacheStatus.IS_NOT)
if set1.cached_is_not_right_functional or set2.cached_is_not_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS_NOT)
elif set1.cached_is_not_relation or set2.cached_is_not_relation:
result.cache_relation(_mo.CacheStatus.IS_NOT)
# Clan flags:
if set1.cached_is_clan and set2.cached_is_clan:
result.cache_clan(_mo.CacheStatus.IS)
if set1.cached_is_absolute and set2.cached_is_absolute:
result.cache_absolute(_mo.CacheStatus.IS)
elif set1.cached_is_not_absolute or set2.cached_is_not_absolute:
result.cache_absolute(_mo.CacheStatus.IS_NOT)
if set1.cached_is_functional and set2.cached_is_functional:
result.cache_functional(_mo.CacheStatus.IS)
elif set1.cached_is_not_functional or set2.cached_is_not_functional:
result.cache_functional(_mo.CacheStatus.IS_NOT)
if set1.cached_is_right_functional and set2.cached_is_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS)
elif set1.cached_is_not_right_functional or set2.cached_is_not_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS_NOT)
if set1.cached_is_not_regular or set2.cached_is_not_regular:
result.cache_regular(_mo.CacheStatus.IS_NOT)
if set1.cached_is_not_right_regular or set2.cached_is_not_right_regular:
result.cache_right_regular(_mo.CacheStatus.IS_NOT)
elif set1.cached_is_not_clan or set2.cached_is_not_clan:
result.cache_clan(_mo.CacheStatus.IS_NOT)
return result
def test_make_or_raise_undef(self):
"""Test make_or_raise_undef() together with RaiseOnUndef."""
try:
self.assertEqual(RaiseOnUndef.get_level(), 0)
self.assertIs(make_or_raise_undef(), Undef())
RaiseOnUndef.set_level(1)
self.assertRaises(UndefException, lambda: make_or_raise_undef())
self.assertIs(make_or_raise_undef(2), Undef())
RaiseOnUndef.set_level(2)
self.assertRaises(UndefException, lambda: make_or_raise_undef(2))
RaiseOnUndef.reset()
self.assertIs(make_or_raise_undef(2), Undef())
except: # Make sure RaiseOnUndef level gets reset.
RaiseOnUndef.reset()
raise
def transpose(multiclan: 'P(P(M x M) x N)',
_checked=True) -> 'P(P(M x M) x N)':
"""Return a multiclan where all relations have their left and right components swapped.
:return: The :term:`unary multi-extension` of :term:`transposition` from the
:term:`algebra of relations` to the :term:`algebra of multiclans`, applied to
``multiclan``, or `Undef()` if ``multiclan`` is not a :term:`multiclan`.
"""
if _checked:
if not is_member(multiclan):
return _undef.make_or_raise_undef2(multiclan)
else:
assert is_member_or_undef(multiclan)
if multiclan is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = _extension.unary_multi_extend(multiclan,
_functools.partial(
_relations.transpose,
_checked=False),
_checked=False)
if not result.is_empty:
result.cache_multiclan(_mo.CacheStatus.IS)
result.cache_absolute(multiclan.cached_absolute)
result.cache_functional(multiclan.cached_right_functional)
result.cache_right_functional(multiclan.cached_functional)
result.cache_reflexive(multiclan.cached_reflexive)
result.cache_symmetric(multiclan.cached_symmetric)
result.cache_transitive(multiclan.cached_transitive)
result.cache_regular(multiclan.cached_right_regular)
result.cache_right_regular(multiclan.cached_regular)
return result
def cross_intersect(multiclan1: 'P(P(M x M) x N)',
multiclan2: 'PP(M x M)',
_checked=True) -> 'PP(M x M)':
r"""Return the :term:`cross-intersection` of ``multiclan1`` and ``multiclan2``.
:return: The :term:`binary multi-extension` of :term:`intersection` from the :term:`algebra
of relations` (which inherits it from the :term:`algebra of sets`) to the :term:`algebra
of multiclans` applied to ``multiclan1`` and ``multiclan2``, or `Undef()` if
``multiclan1`` or ``multiclan2`` are not :term:`multiclan`\s.
"""
if _checked:
if not is_member(multiclan1):
return _undef.make_or_raise_undef2(multiclan1)
if not is_member(multiclan2):
return _undef.make_or_raise_undef2(multiclan2)
else:
assert is_member_or_undef(multiclan1)
assert is_member_or_undef(multiclan2)
if multiclan1 is _undef.Undef() or multiclan2 is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = _extension.binary_multi_extend(multiclan1,
multiclan2,
_functools.partial(
_sets.intersect,
_checked=False),
_checked=False)
if not result.is_empty:
result.cache_multiclan(_mo.CacheStatus.IS)
if multiclan1.cached_is_functional or multiclan2.cached_is_functional:
result.cache_functional(_mo.CacheStatus.IS)
if multiclan1.cached_is_right_functional or multiclan2.cached_is_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS)
return result
def compose(rel1: 'P(M x M)',
rel2: 'P(M x M)',
_checked=True) -> 'P(M x M)':
r"""Return the composition of ``rel1`` with ``rel2``.
:return: The :term:`binary extension` of :term:`composition` from the :term:`algebra of
couplets` to the :term:`algebra of relations`, applied to the :term:`relation`\s
``rel1`` and ``rel2``, or `Undef()` if ``rel1`` or ``rel2`` are not relations.
"""
if _checked:
if not is_member(rel1):
return _undef.make_or_raise_undef2(rel1)
if not is_member(rel2):
return _undef.make_or_raise_undef2(rel2)
else:
assert is_member_or_undef(rel1)
assert is_member_or_undef(rel2)
if rel1 is _undef.Undef() or rel2 is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = _extension.binary_extend(rel1,
rel2,
_functools.partial(_couplets.compose,
_checked=False),
_checked=False)
if not result.is_empty:
result.cache_relation(_mo.CacheStatus.IS)
if rel1.cached_is_absolute and rel2.cached_is_absolute:
result.cache_absolute(_mo.CacheStatus.IS)
if rel1.cached_is_functional and rel2.cached_is_functional:
result.cache_functional(_mo.CacheStatus.IS)
if rel1.cached_is_right_functional and rel2.cached_is_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS)
return result
def demultify(multiset: 'P( M x N )', _checked=True) -> 'P( M )':
"""Return a :term:`set` based on ``multiset`` that contains all elements without multiples."""
if _checked:
if not is_member(multiset):
return _undef.make_or_raise_undef2(multiset)
else:
assert is_member_or_undef(multiset)
if multiset is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = _mo.Set(multiset.data.keys(), direct_load=True)
if not result.is_empty:
result.cache_clan(multiset.cached_multiclan)
if multiset.cached_is_multiclan:
result.cache_absolute(multiset.cached_absolute)
result.cache_functional(multiset.cached_functional)
result.cache_right_functional(multiset.cached_right_functional)
result.cache_reflexive(multiset.cached_reflexive)
result.cache_symmetric(multiset.cached_symmetric)
result.cache_transitive(multiset.cached_transitive)
result.cache_regular(multiset.cached_regular)
result.cache_right_regular(multiset.cached_right_regular)
# We don't yet have a concept of multirelations (multisets of couplets). Because of this,
# a multiset that is converted into a set may be a relation without us being able to know
# this here. Because of this, the only flags we can propagate are multiclan flags.
return result
def intersect(multiset1: 'P( M x N )',
multiset2: 'P( M x N )',
_checked=True) -> 'P( M x N )':
"""Return the multiset intersection of ``multiset1`` with ``multiset2``.
:return: The :term:`multiset intersection` of ``multiset1`` and ``multiset2`` or `Undef()`
if ``multiset1`` or ``multiset2`` are not instances of :class:`~.Multiset`.
"""
if _checked:
if not is_member(multiset1):
return _undef.make_or_raise_undef2(multiset1)
if not is_member(multiset2):
return _undef.make_or_raise_undef2(multiset2)
else:
assert is_member_or_undef(multiset1)
assert is_member_or_undef(multiset2)
if multiset1 is _undef.Undef() or multiset2 is _undef.Undef():
return _undef.make_or_raise_undef(2)
values = multiset1.data & multiset2.data
result = _mo.Multiset(values)
if not result.is_empty:
# Multiclan flags:
if multiset1.cached_is_multiclan or multiset2.cached_is_multiclan:
result.cache_multiclan(CacheStatus.IS)
if multiset1.cached_is_absolute or multiset2.cached_is_absolute:
result.cache_absolute(CacheStatus.IS)
if multiset1.cached_is_functional or multiset2.cached_is_functional:
result.cache_functional(CacheStatus.IS)
if multiset1.cached_is_right_functional or multiset2.cached_is_right_functional:
result.cache_right_functional(CacheStatus.IS)
if multiset1.cached_is_regular or multiset2.cached_is_regular:
result.cache_regular(CacheStatus.IS)
if multiset1.cached_is_right_regular or multiset2.cached_is_right_regular:
result.cache_right_regular(CacheStatus.IS)
return result
def cross_right_functional_union(clan1: 'PP(M x M)',
clan2: 'PP(M x M)',
_checked=True) -> 'PP(M x M)':
r"""Return the cross-right-functional union of ``clan1`` and ``clan2``.
The :term:`cross-right-functional union` of two :term:`clan`\s is the :term:`cross-union`
of these clans, but removing all resulting :term:`relation`\s that are not
:term:`right-functional`.
:return: The :term:`binary extension` of the :term:`right-functional union` from the
:term:`algebra of relations` to the :term:`algebra of clans`, applied to ``clan1`` and
``clan2``, or `Undef()` if ``clan1`` or ``clan2`` are not :term:`clan`\s.
"""
if _checked:
if not is_member(clan1):
return _undef.make_or_raise_undef2(clan1)
if not is_member(clan2):
return _undef.make_or_raise_undef2(clan2)
else:
assert is_member_or_undef(clan1)
assert is_member_or_undef(clan2)
if clan1 is _undef.Undef() or clan2 is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = _extension.binary_extend(
clan1,
clan2,
_functools.partial(_relations.right_functional_union,
_checked=False),
_checked=False)
if not result.is_empty:
result.cache_clan(_mo.CacheStatus.IS)
result.cache_right_functional(_mo.CacheStatus.IS)
if clan1.cached_is_not_functional or clan2.cached_is_not_functional:
result.cache_functional(_mo.CacheStatus.IS_NOT)
return result
def get_lefts(clan: 'PP(M x M)', _checked=True) -> 'P( M )':
r"""Return the set of the left components of all couplets in all relations in ``clan``.
:return: The :term:`union` of the :term:`left set`\s of all :term:`relation`\s in ``clan`` or
`Undef()` if ``clan`` is not a :term:`clan`.
"""
if _checked:
if not is_member(clan):
return _undef.make_or_raise_undef2(clan)
else:
assert is_member_or_undef(clan)
if clan is _undef.Undef():
return _undef.make_or_raise_undef(2)
if clan.is_empty:
# The left set of an empty set is the empty set
return clan
clan_itr = iter(clan)
left_set = _relations.get_lefts(next(clan_itr), _checked=False)
for rel in clan_itr:
left_set = _sets.union(_relations.get_lefts(rel, _checked=False),
left_set,
_checked=False)
if not left_set.is_empty:
if clan.cached_is_absolute:
left_set.cache_absolute(CacheStatus.IS)
return left_set
def cross_intersect(clan1: 'PP(M x M)',
clan2: 'PP(M x M)',
_checked=True) -> 'PP(M x M)':
r"""Return the cross-intersection of ``clan1`` and ``clan2``.
The :term:`cross-intersection` of two :term:`clan`\s is a clan that contains the result of
intersecting every :term:`relation` from one clan with every relation from the other clan.
:return: The :term:`binary extension` of :term:`intersection` from the :term:`algebra of
relations` (which inherits it from the :term:`algebra of sets`) to the :term:`algebra of
clans` applied to ``clan1`` and ``clan2``, or `Undef()` if ``clan1`` or ``clan2`` are
not :term:`clan`\s.
"""
if _checked:
if not is_member(clan1):
return _undef.make_or_raise_undef2(clan1)
if not is_member(clan2):
return _undef.make_or_raise_undef2(clan2)
else:
assert is_member_or_undef(clan1)
assert is_member_or_undef(clan2)
if clan1 is _undef.Undef() or clan2 is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = _extension.binary_extend(clan1,
clan2,
_functools.partial(_sets.intersect,
_checked=False),
_checked=False)
if not result.is_empty:
result.cache_clan(_mo.CacheStatus.IS)
if clan1.cached_is_functional or clan2.cached_is_functional:
result.cache_functional(_mo.CacheStatus.IS)
if clan1.cached_is_right_functional or clan2.cached_is_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS)
return result
def is_superset_of(set1: 'P( M )', set2: 'P( M )', _checked=True) -> bool:
r"""Return whether ``set1`` is a superset of ``set2``.
:return: ``True`` if ``set1`` is a :term:`superset` of ``set2``, ``False`` if not. Return
`Undef()` if ``set1`` or ``set2`` are not :term:`set`\s (that is, instances of
:class:`~.Set`).
"""
if _checked:
if not is_member(set1):
return _undef.make_or_raise_undef()
if not is_member(set2):
return _undef.make_or_raise_undef()
else:
assert is_member(set1)
assert is_member(set2)
return set1.data.issuperset(set2.data)
def get_rights(mclan: 'P(P(M x M) x N)', _checked=True) -> 'P( M )':
r"""Return the set of the right components of all couplets in all relations in ``mclan``.
:return: The :term:`union` of the :term:`right set`\s of all :term:`relation`\s in ``mclan`` or
`Undef()` if ``mclan`` is not a :term:`multiclan`.
"""
if _checked:
if not is_member(mclan):
return _undef.make_or_raise_undef2(mclan)
else:
assert is_member_or_undef(mclan)
if mclan is _undef.Undef():
return _undef.make_or_raise_undef(2)
if mclan.is_empty:
# The right set of an empty set is the empty set
return mclan
clan_itr = iter(mclan)
right_set = _relations.get_rights(next(clan_itr), _checked=False)
for rel in clan_itr:
right_set = _sets.union(_relations.get_rights(rel, _checked=False),
right_set,
_checked=False)
if not right_set.is_empty:
if mclan.cached_is_absolute:
right_set.cache_absolute(_mo.CacheStatus.IS)
return right_set
def is_right_regular(mclan, _checked=True) -> bool:
"""Return whether ``mclan`` is right-regular.
:return: ``True`` if ``mclan`` is :term:`right-regular`, ``False`` if not, or `Undef()` if
``mclan`` is not a :term:`multiclan`.
"""
if _checked:
if not is_member(mclan):
return _undef.make_or_raise_undef2(mclan)
else:
assert is_member_or_undef(mclan)
if mclan is _undef.Undef():
return _undef.make_or_raise_undef(2)
if mclan.cached_right_regular == _mo.CacheStatus.UNKNOWN:
# The empty set is already handled in Set().__init__ via flags initialization.
if mclan.cached_is_not_right_functional:
mclan.cache_right_regular(_mo.CacheStatus.IS_NOT)
return False
itr = iter(mclan.data)
rel = next(itr)
if not _relations.is_right_functional(rel):
mclan.cache_right_regular(_mo.CacheStatus.IS_NOT)
return False
right_set = rel.get_right_set()
right_regular = all(
_relations.is_right_functional(rel)
and right_set == rel.get_right_set() for rel in itr)
mclan.cache_regular(_mo.CacheStatus.from_bool(right_regular))
return mclan.cached_is_regular
def fill_lefts(rel: 'P(M x M)',
renames: 'P(M x M)',
_checked=True) -> 'P(M x M)':
r"""Return the left components in ``rel`` that are missing in ``renames`` as a diagonal
unioned with ``renames``.
The purpose is to create a :term:`relation` that can be used with the :term:`composition`
operation to change (rename) one or more :term:`left component`\s and leave the rest alone.
:param rel: The :term:`relation` that provides the full :term:`left set`.
:param renames: A relation where the :term:`right component`\s are meant to be
:term:`composition` 'origins' and the :term:`left component`\s composition 'targets'.
:return: A relation that contains all members of ``renames`` unioned with a :term:`diagonal`
that consists of all left components in ``rel`` that are missing in ``renames``.
"""
if _checked:
if not is_member(rel):
return _undef.make_or_raise_undef2(rel)
if not is_member(renames):
return _undef.make_or_raise_undef2(renames)
else:
assert is_member_or_undef(rel)
assert is_member_or_undef(renames)
if rel is _undef.Undef() or renames is _undef.Undef():
return _undef.make_or_raise_undef(2)
missing_lefts = _sets.minus(get_lefts(rel, _checked=False),
get_rights(renames, _checked=False),
_checked=False)
diag_missing_lefts = diag(*missing_lefts, _checked=False)
result = _sets.union(renames, diag_missing_lefts, _checked=False)
assert result.cached_is_relation
return result
def get_rights_for_left(mclan: 'P(P(M x M) x N)',
left: '( M )',
_checked=True) -> 'P(M x N)':
"""Return the multiset of the right components of all couplets in the multiclan ``mclan``
associated with the left component ``left``.
:return: The :term:`right multiset` of the :term:`multiclan` ``mclan`` associated with the
:term:`left component` ``left`` or `Undef()` if ``mclan`` is not a multiclan.
"""
if _checked:
if not is_member(mclan):
return _undef.make_or_raise_undef2(mclan)
if left is _undef.Undef():
return _mo.Set()
left = _mo.auto_convert(left)
else:
assert is_member_or_undef(mclan)
assert _mo.is_mathobject_or_undef(left)
if mclan is _undef.Undef():
return _undef.make_or_raise_undef(2)
if left is _undef.Undef():
return _mo.Set()
clan_itr = iter(mclan)
rights = _sets.multify(
_relations.get_rights_for_left(next(clan_itr), left, _checked=False))
for rel in clan_itr:
rights = _multisets.add(_sets.multify(
_relations.get_rights_for_left(rel, left, _checked=False)),
rights,
_checked=False)
if not rights.is_empty:
if mclan.cached_is_absolute:
rights.cache_absolute(_mo.CacheStatus.IS)
return rights
def multify(set_: 'P( M )', _checked=True) -> 'P( M x N )':
"""Return a :term:`multiset` based on ``set_`` where all multiples are set to 1."""
if _checked:
if not is_member(set_):
return _undef.make_or_raise_undef()
else:
assert is_member(set_)
result = _mo.Multiset(set_.data, direct_load=True)
if not result.is_empty:
result.cache_multiclan(set_.cached_clan)
if set_.cached_is_relation:
# We don't yet have a concept of multirelations (multisets of couplets). This would be
# handled here.
pass
elif set_.cached_is_clan:
result.cache_absolute(set_.cached_absolute)
result.cache_functional(set_.cached_functional)
result.cache_right_functional(set_.cached_right_functional)
result.cache_reflexive(set_.cached_reflexive)
result.cache_symmetric(set_.cached_symmetric)
result.cache_transitive(set_.cached_transitive)
result.cache_regular(set_.cached_regular)
result.cache_right_regular(set_.cached_right_regular)
if set_.cached_is_not_relation and set_.cached_is_not_clan:
# set_ is known to be a plain set.
result.cache_absolute(set_.cached_absolute)
result.cache_functional(_mo.CacheStatus.N_A)
result.cache_right_functional(_mo.CacheStatus.N_A)
result.cache_reflexive(_mo.CacheStatus.N_A)
result.cache_symmetric(_mo.CacheStatus.N_A)
result.cache_transitive(_mo.CacheStatus.N_A)
result.cache_regular(_mo.CacheStatus.N_A)
result.cache_right_regular(_mo.CacheStatus.N_A)
return result
def compose(multiclan1: 'P(P(M x M) x N)',
multiclan2: 'P(P(M x M) x N)',
_checked=True) -> 'P(P(M x M) x N)':
r"""Return the composition of ``multiclan1`` with ``multiclan2``.
:return: The :term:`binary multi-extension` of :term:`composition` from the
:term:`algebra of relations` to the :term:`algebra of multiclans`, applied to
``multiclan1`` and ``multiclan2``, or `Undef()` if ``multiclan1`` or ``multiclan2``
are not :term:`multiclan`\s.
"""
if _checked:
if not is_member(multiclan1):
return _undef.make_or_raise_undef2(multiclan1)
if not is_member(multiclan2):
return _undef.make_or_raise_undef2(multiclan2)
else:
assert is_member_or_undef(multiclan1)
assert is_member_or_undef(multiclan2)
if multiclan1 is _undef.Undef() or multiclan2 is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = _extension.binary_multi_extend(multiclan1,
multiclan2,
_functools.partial(
_relations.compose,
_checked=False),
_checked=False)
if not result.is_empty:
result.cache_multiclan(_mo.CacheStatus.IS)
if multiclan1.cached_is_absolute and multiclan2.cached_is_absolute:
result.cache_absolute(_mo.CacheStatus.IS)
if multiclan1.cached_is_functional and multiclan2.cached_is_functional:
result.cache_functional(_mo.CacheStatus.IS)
if multiclan1.cached_is_right_functional and multiclan2.cached_is_right_functional:
result.cache_right_functional(_mo.CacheStatus.IS)
return result
def get_rights_for_left(rel: 'P(M x M)', left: '( M )', _checked=True) -> 'P( M )':
"""Return the set of the right components of all couplets in the relation ``rel`` associated
with the :term:`left component` ``left``.
:return: The :term:`right set` of the :term:`relation` ``rel`` associated with the :term:`left
component` or `Undef()` if ``rel`` is not a :term:`relation`.
"""
if _checked:
if not is_member(rel):
return _undef.make_or_raise_undef2(rel)
if left is _undef.Undef():
return _mo.Set()
left = _mo.auto_convert(left)
else:
assert is_member_or_undef(rel)
assert _mo.is_mathobject_or_undef(left)
if rel is _undef.Undef():
return _undef.make_or_raise_undef(2)
if left is _undef.Undef():
return _mo.Set()
result = _mo.Set((elem.right for elem in rel if elem.left == left), direct_load=True)
if not result.is_empty:
if rel.cached_is_absolute:
result.cache_absolute(_mo.CacheStatus.IS)
return result
def is_regular(clan, _checked=True) -> bool:
"""Return whether ``clan`` is (left-)regular.
:return: ``True`` if ``clan`` is :term:`regular`, ``False`` if not, or `Undef()` if ``clan``
is not a :term:`clan`.
"""
if _checked:
if not is_member(clan):
return _undef.make_or_raise_undef()
else:
assert is_member(clan)
if clan.cached_regular == _mo.CacheStatus.UNKNOWN:
# The empty set is already handled in Set().__init__ via flags initialization.
if clan.cached_is_not_functional:
clan.cache_regular(_mo.CacheStatus.IS_NOT)
return False
itr = iter(clan)
rel = next(itr)
if not _relations.is_functional(rel):
clan.cache_regular(_mo.CacheStatus.IS_NOT)
return False
left_set = rel.get_left_set()
regular = all(
_relations.is_functional(rel) and left_set == rel.get_left_set() for rel in itr)
clan.cache_regular(_mo.CacheStatus.from_bool(regular))
return clan.cached_is_regular
def transpose(rel: 'P(M x M)', _checked=True) -> 'P(M x M)':
"""Return a relation where all couplets have their left and right components swapped.
:return: The :term:`unary extension` of :term:`transposition` from the
:term:`algebra of couplets` to the :term:`algebra of relations`, applied to the
:term:`relation` ``rel``, or `Undef()` if ``rel`` is not a relation.
"""
if _checked:
if not is_member(rel):
return _undef.make_or_raise_undef2(rel)
else:
assert is_member_or_undef(rel)
if rel is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = _extension.unary_extend(rel,
_functools.partial(
_couplets.transpose,
_checked=False),
_checked=False)
if not result.is_empty:
result.cache_relation(_mo.CacheStatus.IS)
result.cache_absolute(rel.cached_absolute)
result.cache_functional(rel.cached_right_functional)
result.cache_right_functional(rel.cached_functional)
result.cache_reflexive(rel.cached_reflexive)
result.cache_symmetric(rel.cached_symmetric)
result.cache_transitive(rel.cached_transitive)
return result
def cross_functional_union(mclan1: 'P(P(M x M) x N)',
mclan2: 'P(P(M x M) x N)',
_checked=True) -> 'P(P(M x M) x N)':
r"""Return the :term:`cross-functional union` of ``mclan1`` and ``mclan2``.
:return: The :term:`binary multi-extension` of the :term:`functional union` from the
:term:`algebra of relations` to the :term:`algebra of multiclans`, applied to
``mclan1`` and ``mclan2``, or `Undef()` if ``mclan1`` or ``mclan2`` are
not :term:`multiclan`\s.
"""
if _checked:
if not is_member(mclan1):
return _undef.make_or_raise_undef2(mclan1)
if not is_member(mclan2):
return _undef.make_or_raise_undef2(mclan2)
else:
assert is_member_or_undef(mclan1)
assert is_member_or_undef(mclan2)
if mclan1 is _undef.Undef() or mclan2 is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = _extension.binary_multi_extend(
mclan1,
mclan2,
_functools.partial(_relations.functional_union, _checked=False),
_checked=False)
if not result.is_empty:
result.cache_multiclan(CacheStatus.IS)
result.cache_functional(CacheStatus.IS)
if mclan1.cached_is_not_right_functional or mclan2.cached_is_not_right_functional:
result.cache_right_functional(CacheStatus.IS_NOT)
return result
def get_rights_for_left(rel: 'P(M x M)',
left: '( M )',
_checked=True) -> 'P( M )':
"""Return the set of the right components of all couplets in the relation ``rel`` associated
with the :term:`left component` ``left``.
:return: The :term:`right set` of the :term:`relation` ``rel`` associated with the :term:`left
component` or `Undef()` if ``rel`` is not a :term:`relation`.
"""
if _checked:
if not is_member(rel):
return _undef.make_or_raise_undef2(rel)
if left is _undef.Undef():
return _mo.Set()
left = _mo.auto_convert(left)
else:
assert is_member_or_undef(rel)
assert _mo.is_mathobject_or_undef(left)
if rel is _undef.Undef():
return _undef.make_or_raise_undef(2)
if left is _undef.Undef():
return _mo.Set()
result = _mo.Set((elem.right for elem in rel if elem.left == left),
direct_load=True)
if not result.is_empty:
if rel.cached_is_absolute:
result.cache_absolute(_mo.CacheStatus.IS)
return result
def unary_multi_extend(multiset: 'P( M x N )', op, _checked=True) -> 'P( M x N )':
r"""Return the :term:`unary extension` of ``op`` from one :term:`algebra` to another algebra.
For this extension, the elements of the extended algebra must be :term:`multiset`\s of the
elements of the original algebra.
:param multiset: A :term:`multiset` with elements on which ``op`` operates.
:param op: A :term:`unary operation` that operates on the elements of ``multiset``.
:return: A set that consists of the defined results of ``op`` when executed on the elements of
``multiset``, or `Undef()` if ``set1`` is not a :class:`~.Multiset`.
"""
if _checked:
if not isinstance(multiset, _mo.Multiset):
return _ud.make_or_raise_undef()
else:
assert multiset.is_multiset
def _get_values(_multiset):
return_count = _collections.Counter()
for elem, multi in _multiset.data.items():
result = op(elem)
if result is not _ud.Undef():
return_count[result] += multi
return return_count
return _mo.Multiset(_get_values(multiset))
def is_equivalence_relation(mo: _mo.MathObject, _checked: bool = True) -> bool:
r"""Return whether ``mo`` is an :term:`equivalence relation` or `Undef()` if not applicable.
Is implemented for :term:`relation`\s, :term:`clan`\s, :term:`multiclan`\s and :term:`set`\s
of (sets of ...) clans. Is also defined (but not yet implemented) for any combination of sets
or :term:`multiset`\s of relations.
"""
if _checked:
if not isinstance(mo, _mo.MathObject):
return _undef.make_or_raise_undef()
# is_reflexive is the only one that is defined for couplets, so it must be evaluated last or it
# may result in erroneous `False` returns.
symmetric = is_symmetric(mo, _checked=False)
if symmetric is _undef.Undef() or not symmetric:
return symmetric
transitive = is_transitive(mo, _checked=False)
if transitive is _undef.Undef() or not transitive:
return transitive
reflexive = is_reflexive(mo, _checked=False)
if reflexive is _undef.Undef() or not reflexive:
return reflexive
return True
def binary_extend(set1: 'P( M )',
set2: 'P( M )',
op,
_checked=True) -> 'P( M )':
r"""Return the :term:`binary extension` of ``op`` from one :term:`algebra` to another algebra.
For this extension, the elements of the extended algebra must be :term:`set`\s of the
elements of the original algebra.
:param set1: A :term:`set` with elements on which ``op`` operates.
:param set2: A set with elements on which ``op`` operates.
:param op: A :term:`binary operation` that operates on the elements of ``set1`` and ``set2``.
:return: A set that consists of the defined results of ``op`` when executed on all combinations
of the elements of ``set1`` and ``set2``, or `Undef()` if either set is not a
:class:`~.Set`.
"""
if _checked:
if not _sets.is_member(set1):
return _undef.make_or_raise_undef2(set1)
if not _sets.is_member(set2):
return _undef.make_or_raise_undef2(set2)
else:
assert _sets.is_member_or_undef(set1)
assert _sets.is_member_or_undef(set2)
if set1 is _undef.Undef() or set2 is _undef.Undef():
return _undef.make_or_raise_undef(2)
def _get_values(_set1, _set2):
for e1 in _set1:
for e2 in _set2:
result = op(e1, e2)
if result is not _undef.Undef():
yield result
return _mo.Set(_get_values(set1, set2), direct_load=True)
def is_right_regular(mclan, _checked=True) -> bool:
"""Return whether ``mclan`` is right-regular.
:return: ``True`` if ``mclan`` is :term:`right-regular`, ``False`` if not, or `Undef()` if
``mclan`` is not a :term:`multiclan`.
"""
if _checked:
if not is_member(mclan):
return _undef.make_or_raise_undef2(mclan)
else:
assert is_member_or_undef(mclan)
if mclan is _undef.Undef():
return _undef.make_or_raise_undef(2)
if mclan.cached_right_regular == _mo.CacheStatus.UNKNOWN:
# The empty set is already handled in Set().__init__ via flags initialization.
if mclan.cached_is_not_right_functional:
mclan.cache_right_regular(_mo.CacheStatus.IS_NOT)
return False
itr = iter(mclan.data)
rel = next(itr)
if not _relations.is_right_functional(rel):
mclan.cache_right_regular(_mo.CacheStatus.IS_NOT)
return False
right_set = rel.get_right_set()
right_regular = all(
_relations.is_right_functional(rel) and right_set == rel.get_right_set() for rel in itr)
mclan.cache_regular(_mo.CacheStatus.from_bool(right_regular))
return mclan.cached_is_regular
def get_rights_for_left(mclan: 'P(P(M x M) x N)', left: '( M )', _checked=True) -> 'P(M x N)':
"""Return the multiset of the right components of all couplets in the multiclan ``mclan``
associated with the left component ``left``.
:return: The :term:`right multiset` of the :term:`multiclan` ``mclan`` associated with the
:term:`left component` ``left`` or `Undef()` if ``mclan`` is not a multiclan.
"""
if _checked:
if not is_member(mclan):
return _undef.make_or_raise_undef2(mclan)
if left is _undef.Undef():
return _mo.Set()
left = _mo.auto_convert(left)
else:
assert is_member_or_undef(mclan)
assert _mo.is_mathobject_or_undef(left)
if mclan is _undef.Undef():
return _undef.make_or_raise_undef(2)
if left is _undef.Undef():
return _mo.Set()
clan_itr = iter(mclan)
rights = _sets.multify(_relations.get_rights_for_left(next(clan_itr), left, _checked=False))
for rel in clan_itr:
rights = _multisets.add(
_sets.multify(_relations.get_rights_for_left(rel, left, _checked=False)),
rights, _checked=False)
if not rights.is_empty:
if mclan.cached_is_absolute:
rights.cache_absolute(_mo.CacheStatus.IS)
return rights
def get_rights(mclan: 'P(P(M x M) x N)', _checked=True) -> 'P( M )':
r"""Return the set of the right components of all couplets in all relations in ``mclan``.
:return: The :term:`union` of the :term:`right set`\s of all :term:`relation`\s in ``mclan`` or
`Undef()` if ``mclan`` is not a :term:`multiclan`.
"""
if _checked:
if not is_member(mclan):
return _undef.make_or_raise_undef2(mclan)
else:
assert is_member_or_undef(mclan)
if mclan is _undef.Undef():
return _undef.make_or_raise_undef(2)
if mclan.is_empty:
# The right set of an empty set is the empty set
return mclan
clan_itr = iter(mclan)
right_set = _relations.get_rights(next(clan_itr), _checked=False)
for rel in clan_itr:
right_set = _sets.union(
_relations.get_rights(rel, _checked=False), right_set, _checked=False)
if not right_set.is_empty:
if mclan.cached_is_absolute:
right_set.cache_absolute(_mo.CacheStatus.IS)
return right_set
def transpose(multiclan: 'P(P(M x M) x N)', _checked=True) -> 'P(P(M x M) x N)':
"""Return a multiclan where all relations have their left and right components swapped.
:return: The :term:`unary multi-extension` of :term:`transposition` from the
:term:`algebra of relations` to the :term:`algebra of multiclans`, applied to
``multiclan``, or `Undef()` if ``multiclan`` is not a :term:`multiclan`.
"""
if _checked:
if not is_member(multiclan):
return _undef.make_or_raise_undef2(multiclan)
else:
assert is_member_or_undef(multiclan)
if multiclan is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = _extension.unary_multi_extend(multiclan, _functools.partial(
_relations.transpose, _checked=False), _checked=False)
if not result.is_empty:
result.cache_multiclan(_mo.CacheStatus.IS)
result.cache_absolute(multiclan.cached_absolute)
result.cache_functional(multiclan.cached_right_functional)
result.cache_right_functional(multiclan.cached_functional)
result.cache_reflexive(multiclan.cached_reflexive)
result.cache_symmetric(multiclan.cached_symmetric)
result.cache_transitive(multiclan.cached_transitive)
result.cache_regular(multiclan.cached_right_regular)
result.cache_right_regular(multiclan.cached_regular)
return result
def transpose(rel: 'P(M x M)', _checked=True) -> 'P(M x M)':
"""Return a relation where all couplets have their left and right components swapped.
:return: The :term:`unary extension` of :term:`transposition` from the
:term:`algebra of couplets` to the :term:`algebra of relations`, applied to the
:term:`relation` ``rel``, or `Undef()` if ``rel`` is not a relation.
"""
if _checked:
if not is_member(rel):
return _undef.make_or_raise_undef2(rel)
else:
assert is_member_or_undef(rel)
if rel is _undef.Undef():
return _undef.make_or_raise_undef(2)
result = _extension.unary_extend(rel, _functools.partial(
_couplets.transpose, _checked=False), _checked=False)
if not result.is_empty:
result.cache_relation(_mo.CacheStatus.IS)
result.cache_absolute(rel.cached_absolute)
result.cache_functional(rel.cached_right_functional)
result.cache_right_functional(rel.cached_functional)
result.cache_reflexive(rel.cached_reflexive)
result.cache_symmetric(rel.cached_symmetric)
result.cache_transitive(rel.cached_transitive)
return result
def is_right_regular(mo: _mo.MathObject, _checked: bool = True) -> bool:
r"""Return whether ``mo`` is :term:`right-regular` or `Undef()` if not applicable.
Is implemented for :term:`clan`\s, :term:`multiclan`\s and :term:`set`\s of (sets of ...) clans.
Is also defined (but not yet implemented) for any combination of sets or :term:`multiset`\s
of clans.
"""
# pylint: disable=too-many-return-statements
if _checked:
if not isinstance(mo, _mo.MathObject):
return _undef.make_or_raise_undef()
# Check cache status.
if mo.cached_right_regular == _mo.CacheStatus.IS:
return True
if mo.cached_right_regular == _mo.CacheStatus.IS_NOT:
return False
if mo.cached_right_regular == _mo.CacheStatus.N_A:
return _undef.make_or_raise_undef(2)
# Check type (right-regular is only defined on Sets and Multisets) and algebra memberships.
if not mo.is_set and not mo.is_multiset:
mo.cache_right_regular(_mo.CacheStatus.N_A)
return _undef.make_or_raise_undef(2)
if _clans.is_member(mo):
return _clans.is_right_regular(mo, _checked=False)
if _multiclans.is_member(mo):
return _multiclans.is_right_regular(mo, _checked=False)
# Check higher (not yet defined) algebras.
if mo.get_ground_set().get_powerset_level(_clans.get_ground_set()) > 0:
mo_iter = iter(mo)
elem1 = next(mo_iter)
if not is_right_regular(elem1):
mo.cache_right_regular(_mo.CacheStatus.IS_NOT)
return False
elem1_rights = elem1.get_rights()
right_regular = all(
is_right_regular(elem, _checked=False)
and elem.get_rights() == elem1_rights for elem in mo_iter)
mo.cache_right_regular(_mo.CacheStatus.from_bool(right_regular))
return mo.cached_is_right_regular
# Nothing applied: 'right-regular' is not defined.
mo.cache_right_regular(_mo.CacheStatus.N_A)
return _undef.make_or_raise_undef(2)
def superstrict(set1: 'P( M )', set2: 'P( M )', _checked=True) -> 'P( M )':
r"""Return ``set1`` if it is a superset of ``set2``, otherwise return `Undef()`.
:return: Return the :term:`superstriction` of ``set1`` and ``set1``; that is, return
``set1`` if it is a :term:`superset` of ``set2`` or `Undef()` if not. Also return
`Undef()` if ``set1`` or ``set2`` are not :term:`set`\s (that is, instances of
:class:`~.Set`).
"""
# pylint: disable=too-many-branches
if _checked:
if not is_member(set1):
return _undef.make_or_raise_undef2(set1)
if not is_member(set2):
return _undef.make_or_raise_undef2(set2)
else:
assert is_member_or_undef(set1)
assert is_member_or_undef(set2)
if set1 is _undef.Undef() or set2 is _undef.Undef():
return _undef.make_or_raise_undef(2)
if not is_superset_of(set1, set2, _checked=False):
return _undef.make_or_raise_undef(2)
if not set1.is_empty:
# Relation flags:
if set1.cached_is_relation:
if set2.cached_is_not_absolute:
set1.cache_absolute(_mo.CacheStatus.IS_NOT)
if set2.cached_is_not_functional:
set1.cache_functional(_mo.CacheStatus.IS_NOT)
if set2.cached_is_not_right_functional:
set1.cache_right_functional(_mo.CacheStatus.IS_NOT)
# Clan flags:
if set1.cached_is_clan:
if set2.cached_is_not_absolute:
set1.cache_absolute(_mo.CacheStatus.IS_NOT)
if set2.cached_is_not_functional:
set1.cache_functional(_mo.CacheStatus.IS_NOT)
if set2.cached_is_not_right_functional:
set1.cache_right_functional(_mo.CacheStatus.IS_NOT)
if set2.cached_is_not_regular:
set1.cache_regular(_mo.CacheStatus.IS_NOT)
if set2.cached_is_not_right_regular:
set1.cache_right_regular(_mo.CacheStatus.IS_NOT)
return set1
def is_reflexive(mo: _mo.MathObject, _checked: bool = True) -> bool:
r"""Return whether ``mo`` is :term:`reflexive` or `Undef()` if not applicable.
Is implemented for :term:`couplet`\s, :term:`relation`\s, :term:`clan`\s, :term:`multiclan`\s
and :term:`set`\s of (sets of ...) clans. Is also defined (but not yet implemented) for any
combination of sets or :term:`multiset`\s of relations.
"""
# pylint: disable=too-many-return-statements
if _checked:
if not isinstance(mo, _mo.MathObject):
return _undef.make_or_raise_undef()
# Check cache status.
if mo.cached_reflexive == _mo.CacheStatus.IS:
return True
if mo.cached_reflexive == _mo.CacheStatus.IS_NOT:
return False
if mo.cached_reflexive == _mo.CacheStatus.N_A:
return _undef.make_or_raise_undef(2)
# Check types and algebra memberships.
if _couplets.is_member(mo):
return _couplets.is_reflexive(mo, _checked=False)
if not mo.is_set and not mo.is_multiset:
mo.cache_reflexive(_mo.CacheStatus.N_A)
return _undef.make_or_raise_undef(2)
if _relations.is_member(mo):
return _relations.is_reflexive(mo, _checked=False)
if _clans.is_member(mo):
return _clans.is_reflexive(mo, _checked=False)
if _multiclans.is_member(mo):
return _multiclans.is_reflexive(mo, _checked=False)
# Check higher (not yet defined) algebras.
reflexive = _is_powerset_property(mo, _clans.get_ground_set(),
is_reflexive)
if reflexive is not _undef.Undef():
mo.cache_reflexive(_mo.CacheStatus.from_bool(reflexive))
return reflexive
# Nothing applied: 'reflexive' is not defined.
mo.cache_reflexive(_mo.CacheStatus.N_A)
return _undef.make_or_raise_undef(2)
def get_right_set(self) -> 'P( M )':
"""Get the :term:`right set` for this :class:`Multiset`. Return `Undef()` if not
applicable.
.. todo:: Once multipowersets are fully implemented,
see :meth:`~algebraixlib.mathobjects.set.Set.get_right_set`.
"""
if _multiclans().is_member(self):
return _multiclans().get_rights(self, _checked=False)
return _ud.make_or_raise_undef()
def __getitem__(self, left) -> 'P( M )':
r"""With the syntax ``mo[left]``, return a set of rights associated with ``left``.
:param left: The :term:`left component` of the :term:`couplet`\(s) of which the
:term:`right component`\(s) are returned.
:return: If ``self`` is a :term:`relation`, return a :term:`set` that contains the
right(s) of the couplet(s) that have a left that matches ``left``. (This set may
be empty if no couplet with the given left exists.) Return `Undef()` if ``self`` is not
a relation.
"""
return _undef.make_or_raise_undef()
