galactic.algebras.connection.core

galactic.algebras.connection.core#

galactic.algebras.connection.core module

Defines classes for working with antitone Galois connections on power sets:

class Polarity(*args, **kwargs)#

Bases: Protocol, Generic[_S, _T]

Represents polarities for antitone Galois connections.

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

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

Note

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

abstractmethod __call__(*elements)#

Compute the correspondence for the given elements.

Parameters:

*elements (Iterable[TypeVar(_S)]) – An optional iterable of elements

Returns:

The corresponding closed set

Return type:

Closed[_T]

Raises:

NotImplementedError – If the method is not implemented

class GaloisConnection#

Bases: Protocol, Generic[_S, _T]

Represents antitone Galois connections.

  • the type _S is the element type of the left side of the antitone Galois connection

  • the type _T is the element type of the right side of the antitone Galois connection

Note

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

Examples

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

Get the polarities of the Galois connection.

Returns:

The pair of polarities

Return type:

tuple[Polarity[_S, _T], Polarity[_T, _S]]

Raises:

NotImplementedError – If the method is not implemented

property closures: tuple[Closure[_S], Closure[_T]]#

Get the closures of the Galois connection.

Returns:

The pair of closure operators

Return type:

tuple[Closure[_S], Closure[_T]

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