🔌 Antitone Galois connection

🔌 Antitone Galois connection#

Classes and functions#

galactic.algebras.connection.core module

Defines classes for working with antitone Galois connections on powersets:

class AbstractPolarity(*args, **kwargs)#

Bases: Protocol[_C, _S, _D]

Represents polarities for antitone Galois connections.

  • the type _C represents the type of the closed of the left part of the antitone Galois connection

  • the type _S represents the type of the elements of the left part of the antitone Galois connection

  • the type _D represents the type of the closed of the right part of the antitone Galois connection

Note

See https://en.wikipedia.org/wiki/Galois_connection#Antitone_Galois_connection.

abstractmethod __call__(*others, elements=None)#

Compute the correspondence for the given closed sets or elements.

Parameters:
Returns:

The corresponding closed set

Return type:

_D

Raises:

NotImplementedError – If the method is not implemented

class AbstractGaloisConnection(*args, **kwargs)#

Bases: Protocol[_C, _S, _D, _T]

Represents antitone Galois connections.

  • the type _C represents the type of the closed of the left part of the antitone Galois connection

  • the type _S represents the type of the elements of the left part of the antitone Galois connection

  • the type _D represents the type of the closed of the right part of the antitone Galois connection

  • the type _T represents the type of the elements of the right part of the antitone Galois connection

Note

See https://en.wikipedia.org/wiki/Galois_connection.

Examples

>>> from galactic.algebras.connection.examples.numerical import IntegerConnection
>>> connection = IntegerConnection(
...     [2, 3, 4, 5, 6],
...     [10, 12, 18, 24, 36, 48, 30, 60],
... )
>>> polarities = connection.polarities
>>> list(polarities[0](elements=[2, 3, 4]))
[12, 24, 36, 48, 60]
>>> list(polarities[1](polarities[0](elements=[2, 3, 4])))
[2, 3, 4, 6]
>>> list(polarities[1]([30]))
[2, 3, 5, 6]
>>> list(polarities[0](polarities[1](elements=[30])))
[30, 60]
>>> list(polarities[1]([10]))
[2, 5]
>>> list(polarities[0](polarities[1](elements=[10])))
[10, 30, 60]
>>> closures = connection.closures
>>> list(closures[0](elements=[2, 3]))
[2, 3, 6]
>>> list(closures[1](elements=[30]))
[30, 60]
abstract property closures: tuple[AbstractClosure[_C, _S], AbstractClosure[_D, _T]]#

Get the closures of the Galois connection.

Returns:

The pair of closure operators

Return type:

tuple[AbstractClosure[_C, _S], AbstractClosure[_D, _T]]

Raises:

NotImplementedError – If the method is not implemented

abstract property polarities: tuple[AbstractPolarity[_C, _S, _D], AbstractPolarity[_D, _T, _C]]#

Get the polarities of the Galois connection.

Returns:

The pair of polarities

Return type:

tuple[AbstractPolarity[_C, _S, _D], AbstractPolarity[_D, _T, _C]]

Raises:

NotImplementedError – If the method is not implemented

class GaloisConnectionMixin#

Bases: Generic[_C, _S, _D, _T]

Represents antitone Galois connections.

  • the type _C represents the type of the closed of the left part of the antitone Galois connection

  • the type _S represents the type of the elements of the left part of the antitone Galois connection

  • the type _D represents the type of the closed of the right part of the antitone Galois connection

  • the type _T represents the type of the elements of the right part of the antitone Galois connection

Notes

See https://en.wikipedia.org/wiki/Galois_connection.

Examples

>>> from galactic.algebras.connection.examples.numerical import IntegerConnection
>>> connection = IntegerConnection(
...     [2, 3, 4, 5, 6],
...     [10, 12, 18, 24, 36, 48, 30, 60],
... )
>>> polarities = connection.polarities
>>> list(polarities[0](elements=[2, 3, 4]))
[12, 24, 36, 48, 60]
>>> list(polarities[1](polarities[0](elements=[2, 3, 4])))
[2, 3, 4, 6]
>>> list(polarities[1]([30]))
[2, 3, 5, 6]
>>> list(polarities[0](polarities[1](elements=[30])))
[30, 60]
>>> list(polarities[1]([10]))
[2, 5]
>>> list(polarities[0](polarities[1](elements=[10])))
[10, 30, 60]
>>> closures = connection.closures
>>> list(closures[0](elements=[2, 3]))
[2, 3, 6]
>>> list(closures[1](elements=[30]))
[30, 60]
property closures: tuple[AbstractClosure[_C, _S], AbstractClosure[_D, _T]]#

Get the closures of the Galois connection.

Returns:

The pair of closure operators

Return type:

tuple[AbstractClosure[_C, _S], AbstractClosure[_D, _T]]

Examples#

galactic.algebras.connection.examples.numerical module

Defines example classes for working with antitone finite Galois connections:

class IntegerConnection(domain, co_domain)#

Bases: GaloisConnectionMixin[Divisors, int, Multiples, int]

Represents an antitone Galois connection between divisors and multiples.

Parameters:

  • domain (Iterable[int]) – An iterable of integers for the domain

  • co_domain (Iterable[int]) – An iterable of integers for the co-domain

property closures: tuple[AbstractClosure[_C, _S], AbstractClosure[_D, _T]]#

Get the closures of the Galois connection.

Returns:

The pair of closure operators

Return type:

tuple[AbstractClosure[_C, _S], AbstractClosure[_D, _T]]

property co_domain: FrozenFIFOSet[int]#

Get the co-domain.

Returns:

The co-domain

Return type:

FrozenFIFOSet[int]

property domain: FrozenFIFOSet[int]#

Get the domain.

Returns:

The domain

Return type:

FrozenFIFOSet[int]

property polarities: tuple[MultiplePolarity, DivisorPolarity]#

Get the polarities.

Returns:

The couple of polarities

Return type:

tuple[MultiplePolarity, DivisorPolarity]

class DivisorPolarity(connection)#

Bases: object

Represents the polarity for divisors.

__call__(*others, elements=None)#

Compute the correspondence of elements.

Parameters:

  • *others (Multiples) – A sequence of Multiple

  • elements (Iterable[int] | None, default: None) – An optional iterable of integers

Returns:

The common divisors of elements

Return type:

Divisors

class MultiplePolarity(connection)#

Bases: object

Represents the polarity for multiples.

Parameters:

connection (IntegerConnection) – The antitone Galois connection

__call__(*others, elements=None)#

Compute the correspondence of elements.

Parameters:

  • *others (Divisors) – A sequence of divisors

  • elements (Iterable[int] | None, default: None) – An optional iterable of integers

Returns:

The common multiples of elements

Return type:

Multiples

class Divisors(connection, elements=None)#

Bases: ClosedEnumerableMixin[int], Element

Represents the set of divisors of given integers.

Parameters:
Raises:

ValueError – If an element is not in the co-domain

property closure: AbstractClosure[Divisors, int]#

Get the closure operator.

Returns:

The closure operator

Return type:

AbstractClosure[Divisors, int]

equivalence(element)#

Compute the equivalence elements of the given element.

Parameters:

element (TypeVar(_T)) – The element for which equivalence elements are requested

Returns:

An iterator over the equivalence elements of the element

Return type:

Iterator[_T]

join(*others)#

Return the supremum of this element and the given elements.

Parameters:

*others (Self) – The other elements

Returns:

The join of this element and the others

Return type:

Self

join_level(generators)#

Compute the level of this element with respect to an iterable of generators.

Parameters:

generators (Iterable[Self]) – An iterable of generators

Returns:

The level of the element

Return type:

int

join_priority(generators)#

Compute the priority of this element with respect to an iterable of generators.

Parameters:

generators (Iterable[Self]) – An iterable of generators

Returns:

The priority

Return type:

int

meet(*others)#

Return the infimum of this element and the others.

Parameters:

*others (Self) – The other elements

Returns:

The meet of this element and the others

Return type:

Self

meet_level(generators)#

Compute the level of this element with respect to an iterable of generators.

Parameters:

generators (Iterable[Self]) – An iterable of generators

Returns:

The level of the element

Return type:

int

meet_priority(generators)#

Compute the priority of this element with respect to an iterable of generators.

Parameters:

generators (Iterable[Self]) – An iterable of generators

Returns:

The priority

Return type:

int

subsumption(element)#

Compute the elements subsumed by the given element.

Parameters:

element (TypeVar(_T)) – The element for which subsumed elements are requested

Returns:

An iterator over the subsumed elements of the element

Return type:

Iterator[_T]

property support: float#

Compute the support of the closed set.

Returns:

The support of the closed set

Return type:

float

supsumption(element)#

Compute the elements supsumed by the given element.

Notes

Supsumption is a neologism designating the inverse relation of subsumption.

Parameters:

element (TypeVar(_T)) – The element for which supsumed elements are requested

Returns:

An iterator over the supsumed elements of the element

Return type:

Iterator[_T]

class Multiples(connection, elements=None)#

Bases: ClosedEnumerableMixin[int], Element

Represents the set of multiples of given integers.

Parameters:
Raises:

ValueError – If an element is not in the domain

property closure: AbstractClosure[Multiples, int]#

Get the closure operator.

Returns:

The closure operator

Return type:

AbstractClosure[Multiples, int]

equivalence(element)#

Compute the equivalence elements of the given element.

Parameters:

element (TypeVar(_T)) – The element for which equivalence elements are requested

Returns:

An iterator over the equivalence elements of the element

Return type:

Iterator[_T]

join(*others)#

Return the supremum of this element and the given elements.

Parameters:

*others (Self) – The other elements

Returns:

The join of this element and the others

Return type:

Self

join_level(generators)#

Compute the level of this element with respect to an iterable of generators.

Parameters:

generators (Iterable[Self]) – An iterable of generators

Returns:

The level of the element

Return type:

int

join_priority(generators)#

Compute the priority of this element with respect to an iterable of generators.

Parameters:

generators (Iterable[Self]) – An iterable of generators

Returns:

The priority

Return type:

int

meet(*others)#

Return the infimum of this element and the others.

Parameters:

*others (Self) – The other elements

Returns:

The meet of this element and the others

Return type:

Self

meet_level(generators)#

Compute the level of this element with respect to an iterable of generators.

Parameters:

generators (Iterable[Self]) – An iterable of generators

Returns:

The level of the element

Return type:

int

meet_priority(generators)#

Compute the priority of this element with respect to an iterable of generators.

Parameters:

generators (Iterable[Self]) – An iterable of generators

Returns:

The priority

Return type:

int

subsumption(element)#

Compute the elements subsumed by the given element.

Parameters:

element (TypeVar(_T)) – The element for which subsumed elements are requested

Returns:

An iterator over the subsumed elements of the element

Return type:

Iterator[_T]

property support: float#

Compute the support of the closed set.

Returns:

The support of the closed set

Return type:

float

supsumption(element)#

Compute the elements supsumed by the given element.

Notes

Supsumption is a neologism designating the inverse relation of subsumption.

Parameters:

element (TypeVar(_T)) – The element for which supsumed elements are requested

Returns:

An iterator over the supsumed elements of the element

Return type:

Iterator[_T]

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