🔌 Antitone Galois connection

Classes and functions

galactic.algebras.connection.core module

It defines several classes for dealing with antitone Galois connection defined on powersets:

• AbstractPolarity

• AbstractGaloisConnection

• GaloisConnectionMixin

---

class AbstractPolarity(*args, **kwargs)

Bases: Protocol[_C, _S, _D]

It 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)

Call this instance.

Parameters:

• *others (TypeVar(_C, bound= AbstractClosed[Any])) – A sequence of AbstractClosed.

• elements (Iterable[TypeVar(_S)] | None, default: None) – An optional iterable of elements.

Returns:

The correspondence of the arguments.

Return type:

_D

Raises:

NotImplementedError –

class AbstractGaloisConnection(*args, **kwargs)

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

It 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.

Returns:

The closures.

Return type:

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

Raises:

NotImplementedError –

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

Get the polarities.

Returns:

The polarities.

Return type:

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

Raises:

NotImplementedError –

class GaloisConnectionMixin

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

It 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]

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

Get the closures.

Returns:

The closures.

Return type:

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

Examples

galactic.algebras.connection.examples.numerical module

It defines example classes for dealing with antitone finite Galois connection:

• IntegerConnection

• MultiplePolarity

• DivisorPolarity

• Multiple

• Divisors

---

class IntegerConnection(domain, co_domain)

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

It represents an antitone Galois connection.

Parameters:

• domain (Iterable[int]) – An iterable of int.

• co_domain (Iterable[int]) – An iterable of int.

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

Get the closures.

Returns:

The closures.

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

It represents the polarity of divisors.

__call__(*others, elements=None)

Compute the correspondence of elements.

Parameters:

• *others (Multiple) – A sequence of Multiple.

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

Returns:

The common divisors of elements.

Return type:

Divisors

class MultiplePolarity(connection)

Bases: object

It represents the polarity of multiple.

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 int.

Returns:

The common multiple of elements.

Return type:

Multiple

class Divisors(connection, elements=None)

Bases: ClosedEnumerableMixin[int], Element

It represents the divisors of some integers.

Parameters:

• connection (IntegerConnection) – An antitone Galois connection.

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

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 equivalences of the element.

Parameters:

element (TypeVar(_T)) – The element whose equivalences are requested

Returns:

An iterator over the equivalence elements of element

Return type:

Iterator[_T]

join(*others)

Return the supremum of this element and the others.

Parameters:

*others (Self) – The others elements

Returns:

The join of this element and the others

Return type:

Self

join_level(generators)

Compute the level of an element considering 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 an element considering 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 others elements

Returns:

The meet of this element and the others

Return type:

Self

meet_level(generators)

Compute the level of an element considering 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 an element considering an iterable of generators.

Parameters:

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

Returns:

The priority.

Return type:

int

subsumption(element)

Compute subsumed elements of the element.

Parameters:

element (TypeVar(_T)) – The element whose subsumed elements are requested

Returns:

An iterator over the subsumed elements of element

Return type:

Iterator[_T]

property support: float

Get the closed set support.

Returns:

The closed set support.

Return type:

float

supsumption(element)

Compute supsumed elements of the element.

Notes

supsumption is a neologism to designate the inverse relation of subsumption.

Parameters:

element (TypeVar(_T)) – The element whose supsumed elements are requested

Returns:

An iterator over the supsumed elements of element

Return type:

Iterator[_T]

class Multiple(connection, elements=None)

Bases: ClosedEnumerableMixin[int], Element

It represents the multiple of some integers.

Parameters:

• connection (IntegerConnection) – The Galois connection

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

Raises:

ValueError – If an element is not in the domain.

property closure: AbstractClosure[Multiple, int]

Get the closure operator.

Returns:

The closure operator.

Return type:

AbstractClosure[Multiple, int]

equivalence(element)

Compute equivalences of the element.

Parameters:

element (TypeVar(_T)) – The element whose equivalences are requested

Returns:

An iterator over the equivalence elements of element

Return type:

Iterator[_T]

join(*others)

Return the supremum of this element and the others.

Parameters:

*others (Self) – The others elements

Returns:

The join of this element and the others

Return type:

Self

join_level(generators)

Compute the level of an element considering 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 an element considering 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 others elements

Returns:

The meet of this element and the others

Return type:

Self

meet_level(generators)

Compute the level of an element considering 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 an element considering an iterable of generators.

Parameters:

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

Returns:

The priority.

Return type:

int

subsumption(element)

Compute subsumed elements of the element.

Parameters:

element (TypeVar(_T)) – The element whose subsumed elements are requested

Returns:

An iterator over the subsumed elements of element

Return type:

Iterator[_T]

property support: float

Get the closed set support.

Returns:

The closed set support.

Return type:

float

supsumption(element)

Compute supsumed elements of the element.

Notes

supsumption is a neologism to designate the inverse relation of subsumption.

Parameters:

element (TypeVar(_T)) – The element whose supsumed elements are requested

Returns:

An iterator over the supsumed elements of element

Return type:

Iterator[_T]