Antitone Galois connection

The galactic.algebras.connection module.

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

AbstractPolarity;
AbstractPolarityEnumerable;
AbstractGaloisConnection;
AbstractGaloisConnectionLeftEnumerable;
AbstractGaloisConnectionRightEnumerable;
AbstractGaloisConnectionEnumerable;
GaloisConnectionMixin;
GaloisConnectionLeftEnumerableMixin;
GaloisConnectionRightEnumerableMixin;
GaloisConnectionEnumerableMixin;

class AbstractPolarity(*args, **kwargs)

Bases: Protocol[_S, _T]

It represents polarities for antitone Galois connection.

the type _S represents the type of the elements of the left part of the Galois connection;
the type _T represents the type of the elements of the right part of the Galois connection.

Note

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

abstract __call__(*others, elements=None)

Call this instance.

Parameters:

*others (AbstractClosed[TypeVar(_S)]) – A sequence of AbstractClosed.
elements (Iterable[TypeVar(_S)] | None, default: None) – An optional iterable of elements.

Returns:

The correspondence of the arguments.

Return type:

AbstractClosed[_T]

Raises:

NotImplementedError –

class AbstractPolarityEnumerable(*args, **kwargs)

Bases: Protocol[_S, _T]

It represents enumerable polarities for antitone Galois connection.

the type _S represents the type of the elements of the left part of the Galois connection;
the type _T represents the type of the elements of the right part of the Galois connection.

Note

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

abstract __call__(*others, elements=None)

Call this instance.

Parameters:

*others (AbstractClosed[TypeVar(_S)]) – A sequence of AbstractClosed.
elements (Iterable[TypeVar(_S)] | None, default: None) – An optional iterable of elements.

Returns:

The correspondence of the arguments.

Return type:

AbstractClosedEnumerable[_T]

Raises:

NotImplementedError –

class AbstractGaloisConnection(*args, **kwargs)

Bases: Protocol[_S, _T]

It represents antitone Galois connection.

the type _S represents the type of the elements of the left part of the Galois connection;
the type _T represents the type of the elements of the right part of the Galois connection.

Note

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

abstract property polarities: tuple[AbstractPolarity[_S, _T], AbstractPolarity[_T, _S]]

Get the polarities.

Returns:: The polarities.
Return type:: tuple[AbstractPolarity[_S, _T], AbstractPolarity[_T, _S]]
Raises:: NotImplementedError –

abstract property closures: tuple[AbstractClosure[_S], AbstractClosure[_T]]

Get the closures.

Returns:: The closures.
Return type:: tuple[AbstractClosure[_S], AbstractClosure[_T]]
Raises:: NotImplementedError –

class AbstractGaloisConnectionLeftEnumerable(*args, **kwargs)

Bases: Protocol[_S, _T]

It represents antitone Galois connection.

the type _S represents the type of the elements of the left part of the Galois connection;
the type _T represents the type of the elements of the right part of the Galois connection.

Note

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

abstract property polarities: tuple[AbstractPolarity[_S, _T], AbstractPolarityEnumerable[_T, _S]]

Get the polarities.

Returns:: The polarities.
Return type:: tuple[AbstractPolarity[_S, _T], AbstractPolarityEnumerable[_T, _S]]
Raises:: NotImplementedError –

abstract property closures: tuple[AbstractClosureEnumerable[_S], AbstractClosure[_T]]

Get the closures.

Returns:: The closures.
Return type:: tuple[AbstractClosureEnumerable[_S], AbstractClosure[_T]]
Raises:: NotImplementedError –

class AbstractGaloisConnectionRightEnumerable(*args, **kwargs)

Bases: Protocol[_S, _T]

It represents antitone Galois connection.

the type _S represents the type of the elements of the left part of the Galois connection;
the type _T represents the type of the elements of the right part of the Galois connection.

Note

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

abstract property polarities: tuple[AbstractPolarityEnumerable[_S, _T], AbstractPolarity[_T, _S]]

Get the polarities.

Returns:: The polarities.
Return type:: tuple[AbstractPolarityEnumerable[_S, _T], AbstractPolarity[_T, _S]]
Raises:: NotImplementedError –

abstract property closures: tuple[AbstractClosure[_S], AbstractClosureEnumerable[_T]]

Get the closures.

Returns:: The closures.
Return type:: tuple[AbstractClosure[_S], AbstractClosureEnumerable[_T]]
Raises:: NotImplementedError –

class AbstractGaloisConnectionEnumerable(*args, **kwargs)

Bases: Protocol[_S, _T]

It represents antitone Galois connection.

the type _S represents the type of the elements of the left part of the Galois connection;
the type _T represents the type of the elements of the right part of the Galois connection.

Note

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

Examples

>>> from galactic.algebras.examples.connection 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 polarities: tuple[AbstractPolarityEnumerable[_S, _T], AbstractPolarityEnumerable[_T, _S]]

Get the polarities.

Returns:: The polarities.
Return type:: tuple[AbstractPolarityEnumerable[_S, _T], AbstractPolarityEnumerable[_T, _S]]
Raises:: NotImplementedError –

abstract property closures: tuple[AbstractClosureEnumerable[_S], AbstractClosureEnumerable[_T]]

Get the closures.

Returns:: The closures.
Return type:: tuple[AbstractClosureEnumerable[_S], AbstractClosureEnumerable[_T]]
Raises:: NotImplementedError –

class GaloisConnectionMixin

Bases: Generic[_S, _T]

It represents antitone Galois connection.

the type _S represents the type of the elements of the left part of the Galois connection;
the type _T represents the type of the elements of the right part of the Galois connection.

Note

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

Examples

>>> from galactic.algebras.examples.connection 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[_S], AbstractClosure[_T]]

Get the closures.

Returns:: The closures.
Return type:: tuple[AbstractClosure[_S], AbstractClosure[_T]]

class GaloisConnectionLeftEnumerableMixin

Bases: GaloisConnectionMixin[_S, _T], Generic[_S, _T]

It represents left enumerable antitone Galois connection.

the type _S represents the type of the elements of the left part of the Galois connection;
the type _T represents the type of the elements of the right part of the Galois connection.

Note

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

property closures: tuple[AbstractClosureEnumerable[_S], AbstractClosure[_T]]

Get the closures.

Returns:: The closures.
Return type:: tuple[AbstractClosureEnumerable[_S], AbstractClosure[_T]]

class GaloisConnectionRightEnumerableMixin

Bases: GaloisConnectionMixin[_S, _T], Generic[_S, _T]

It represents right enumerable antitone Galois connection.

the type _S represents the type of the elements of the left part of the Galois connection;
the type _T represents the type of the elements of the right part of the Galois connection.

Note

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

property closures: tuple[AbstractClosureEnumerable[_S], AbstractClosure[_T]]

Get the closures.

Returns:: The closures.
Return type:: tuple[AbstractClosureEnumerable[_S], AbstractClosure[_T]]

class GaloisConnectionEnumerableMixin

Bases: GaloisConnectionMixin[_S, _T], Generic[_S, _T]

It represents enumerable antitone Galois connection.

the type _S represents the type of the elements of the left part of the Galois connection;
the type _T represents the type of the elements of the right part of the Galois connection.

Note

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

property closures: tuple[AbstractClosureEnumerable[_S], AbstractClosureEnumerable[_T]]

Get the closures.

Returns:: The closures.
Return type:: tuple[AbstractClosureEnumerable[_S], AbstractClosureEnumerable[_T]]