Elements¶

The galactic.algebras.poset.elements module defines essential type for representing partially ordered elements.

class galactic.algebras.poset.elements.PartiallyOrdered¶

Bases: collections.abc.Hashable

A partially ordered type sets for each pair of elements \(a\), \(b\) either:

\(a \leq b\)
\(b \leq a\)
a and b are incomparable

The relation \(\leq\) must be:

reflexive (\(a \leq a\))
transitive (\(a \leq b\) and \(b \leq c\) implies \(a \leq c\))
anti-symmetric (\(a \leq b\) and \(b \leq a\) implies \(a = b\))

A class implementing the PartiallyOrdered abstract class must be declared by inheriting from PartiallyOrdered with itself as a parameter and must implement the methods:

__lt__()
__le__()
__gt__()
__ge__()
__eq__()
__hash__()

Example

Let the integers ordered by the relation \(a \leq b\): is \(a\) a divisor of \(b\)?

implemented by the Integer class:

>>> from galactic.examples.arithmetic.algebras import Integer
>>> Integer(5) <= Integer(10)
True
>>> Integer(5) <= Integer(11)
False
>>>

abstract __lt__(other)¶

Test if this element is lesser than the other.

Parameters

other – the other element

Returns

bool – True if this element is lesser than the other.
NotImplementedType – if the operation is not implemented between the two objects

abstract __gt__(other)¶

Test if this element is greater than the other.

Parameters

other – the other element

Returns

bool – True if this element is greater than the other.
NotImplementedType – if the operation is not implemented between the two objects

abstract __eq__(other)¶

Test if this element is equal to the other.

Parameters

other – the other element

Returns

bool – True if this element is equal to the other.
NotImplementedType – if the operation is not implemented between the two objects

abstract __le__(other)¶

Test if this element is lesser than or equal to the other.

Parameters

other – the other element

Returns

bool – True if this element is lesser than or equal to the other.
NotImplementedType – if the operation is not implemented between the two objects

__ge__(other)¶

Test if this element is greater than or equal to the other.

Parameters

other – the other element

Returns

bool – True if this element is greater than or equal to the other.
NotImplementedType – if the operation is not implemented between the two objects

class galactic.algebras.poset.elements.UpperBounded¶

Bases: galactic.algebras.poset.elements.PartiallyOrdered

The UpperBounded type defines a maximum value for instances.

abstract classmethod maximum() → galactic.algebras.poset.elements.UpperBounded¶

Returns the maximum element of this type.

Returns: the maximum element
Return type: UpperBounded

class galactic.algebras.poset.elements.LowerBounded¶

Bases: galactic.algebras.poset.elements.PartiallyOrdered

The LowerBounded type defines a minimum value for instances.

abstract classmethod minimum() → galactic.algebras.poset.elements.LowerBounded¶

Returns the minimum element of this type

Returns: the minimum element
Return type: LowerBounded

galactic.algebras.poset.elements.top(iterable: Iterable[galactic.algebras.poset.elements.PartiallyOrdered] = None) → Iterator[galactic.algebras.poset.elements.PartiallyOrdered]¶

The top() function computes an iterator over the top elements of the iterable collection. This operation computes in \(O(n^2)\) where \(n\) is the number of elements in the iterable.

Keyword Arguments: iterable (Iterable[PartiallyOrdered]) – An iterable of partially ordered elements.
Returns: An iterator over the top elements.
Return type: Iterator[PartiallyOrdered]

Example

>>> from galactic.algebras.poset.elements import top
>>> values = [{1, 2, 3}, {1, 2, 3, 4}, {2, 3, 4, 5}]
>>> list(top(iterable=[frozenset(value) for value in values]))
[frozenset({1, 2, 3, 4}), frozenset({2, 3, 4, 5})]

galactic.algebras.poset.elements.bottom(iterable: Iterable[galactic.algebras.poset.elements.PartiallyOrdered] = None) → Iterator[galactic.algebras.poset.elements.PartiallyOrdered]¶

The bottom() function computes an iterator over the bottom elements of the iterable collection. This operation computes in \(O(n^2)\) where \(n\) is the number of elements in the iterable.

Keyword Arguments: iterable (Iterable[PartiallyOrdered]) – An iterable of partially ordered elements.
Returns: An iterator over the bottom elements.
Return type: Iterator[PartiallyOrdered]

Example

>>> from galactic.algebras.poset.elements import top
>>> values = [{1, 2, 3}, {1, 2, 3, 4}, {2, 3, 4, 5}]
>>> list(bottom(iterable=[frozenset(value) for value in values]))
[frozenset({1, 2, 3}), frozenset({2, 3, 4, 5})]

galactic.algebras.poset.elements.upper_limit(iterable: Iterable[galactic.algebras.poset.elements.PartiallyOrdered] = None, limit: galactic.algebras.poset.elements.PartiallyOrdered = None, strict: bool = False) → Iterator[galactic.algebras.poset.elements.PartiallyOrdered]¶

The upper_limit() function computes an iterator over the elements lesser than the limit. This operation computes in \(O(n)\) where \(n\) is the number of elements in the iterable.

Keyword Arguments

iterable (Iterable[PartiallyOrdered]) – An iterable of partially ordered elements.
limit – The upper limit
strict (bool) – Is the comparison strict?

Returns

An iterator over the selected elements.

Return type

Iterator[PartiallyOrdered]

Example

>>> from galactic.algebras.poset.elements import upper_limit
>>> list(upper_limit(iterable=[{1, 2}, {1, 2, 3}, {2, 3}], limit={2, 3}))
[{2, 3}]

galactic.algebras.poset.elements.lower_limit(iterable: Iterable[galactic.algebras.poset.elements.PartiallyOrdered] = None, limit: galactic.algebras.poset.elements.PartiallyOrdered = None, strict: bool = False) → Iterator[galactic.algebras.poset.elements.PartiallyOrdered]¶

The lower_limit() function computes an iterator over the elements greater than the limit. This operation computes in \(O(n)\) where \(n\) is the number of elements in the iterable.

Keyword Arguments

iterable (Iterable[PartiallyOrdered]) – An iterable of partially ordered elements.
limit – The lower limit
strict (bool) – Is the comparison strict?

Returns

An iterator over the selected elements.

Return type

Iterator[PartiallyOrdered]

Example

>>> from galactic.algebras.poset.elements import lower_limit
>>> list(lower_limit(iterable=[{1, 2}, {1, 2, 3}, {2, 3}], limit={2, 3}))
[{1, 2, 3}, {2, 3}]