📘 Antitone Galois connections

Antitone Galois connections are a fundamental concept in order theory and lattice theory, establishing a relationship between two partially ordered sets (posets) through a pair of antitone functions that are adjoint to each other.

In the GALACTIC framework, antitone Galois connections are implemented using specific classes and methods.

Closure operators

The GaloisConnection class from the galactic.algebras.concept.core module represents an antitone Galois connection between two posets (parts of two universes). It provides methods to create and manipulate antitone Galois connections, as well as to compute the associated closure operators.

from galactic.algebras.concept.core import (
    GaloisConnection,
    create_context_from_dataset
)
from galactic.algebras.concept.examples.animals.core import ANIMAL_ATTRS, ANIMAL_DATA
context = create_context_from_dataset(ANIMAL_DATA, ANIMAL_ATTRS)
connection = GaloisConnection(context)
display(connection, connection.polarities, connection.closures, connection.context)

<galactic.algebras.concept.core.GaloisConnection object at 0x72ef32f66940>

(<galactic.algebras.concept.core.AttrPolarity object at 0x72ef32f6c3d0>,
 <galactic.algebras.concept.core.ItemPolarity object at 0x72ef32f6c400>)

(compose(<galactic.algebras.concept.core.ItemPolarity object at 0x72ef32f6c400>, <galactic.algebras.concept.core.AttrPolarity object at 0x72ef32f6c3d0>),
 compose(<galactic.algebras.concept.core.AttrPolarity object at 0x72ef32f6c3d0>, <galactic.algebras.concept.core.ItemPolarity object at 0x72ef32f6c400>))

<galactic.algebras.concept.core.Context object at 0x72ef511e17b0>

The closures operators can be accessed using the closures attribute of the GaloisConnection instance. These operators allow for the computation of the closure of a collection of items or attributes within the context of the antitone Galois connection.

item_closure = connection.closures[0]
attr_closure = connection.closures[1]
display(item_closure, attr_closure)
closed_items = item_closure(
    [
        context.domain.item(key='Cat'),
        context.domain.item(key='Dog'),
    ],
)
display(closed_items, list(closed_items), len(closed_items))
closed_attrs = attr_closure(
    [
        context.co_domain.attr(name='feathers'),
    ],
)
display(closed_attrs, list(closed_attrs), len(closed_attrs))

compose(<galactic.algebras.concept.core.ItemPolarity object at 0x72ef32f6c400>, <galactic.algebras.concept.core.AttrPolarity object at 0x72ef32f6c3d0>)

compose(<galactic.algebras.concept.core.AttrPolarity object at 0x72ef32f6c3d0>, <galactic.algebras.concept.core.ItemPolarity object at 0x72ef32f6c400>)

<galactic.algebras.concept.core.Extent object at 0x72ef4c1eb880>

[Item(key='Fox', value=<list object at 0x72ef32f64900>),
 Item(key='Dog', value=<list object at 0x72ef32f64940>),
 Item(key='Wolf', value=<list object at 0x72ef32f64980>),
 Item(key='Cat', value=<list object at 0x72ef32f649c0>),
 Item(key='Tiger', value=<list object at 0x72ef32f64a00>),
 Item(key='Lion', value=<list object at 0x72ef32f64a40>),
 Item(key='Horse', value=<list object at 0x72ef32f64a80>),
 Item(key='Zebra', value=<list object at 0x72ef32f64ac0>)]

8

<galactic.algebras.concept.core.Intent object at 0x72ef32f84680>

[<function twolegs at 0x72ef32f55da0>, <function feathers at 0x72ef32f55580>]

2

Closed sets of items and attributes

The application of the closure operators demonstrates how the antitone Galois connection can be used to derive closed sets of items and attributes based on the relationships defined in the context. The first closure operation computes the closure of the items “Cat” and “Dog”, resulting in a closed set that includes all items sharing the same attributes using the Extent class. The second closure operation computes the closure of the attribute “feathers”, resulting in a closed set of attributes that are common to all items possessing that attribute using the Intent class.

The Extent and Intent classes implement the Closed protocol from the galactic.algebras.closure.core module, providing a structured way to represent closed sets of items and attributes within the framework of antitone Galois connections.

from galactic.algebras.concept.core import Intent, Extent
extent_1 = Extent(
    connection,
    [context.co_domain.attr(name='feathers')],
)
extent_2 = Extent(
    connection,
    [context.co_domain.attr(name='medium')],
)
join = extent_1 | extent_2
meet = extent_1 & extent_2
display(extent_1, list(extent_1), len(extent_1))
display(extent_2, list(extent_2), len(extent_2))
display(join, list(join), len(join))
display(meet, list(meet), len(meet))

<galactic.algebras.concept.core.Extent object at 0x72ef32f86100>

[Item(key='Dove', value=<list object at 0x72ef32f4c700>),
 Item(key='Hen', value=<list object at 0x72ef32f647c0>),
 Item(key='Duck', value=<list object at 0x72ef32f64500>),
 Item(key='Goose', value=<list object at 0x72ef32f64800>),
 Item(key='Owl', value=<list object at 0x72ef32f64840>),
 Item(key='Hawk', value=<list object at 0x72ef32f64880>),
 Item(key='Eagle', value=<list object at 0x72ef32f648c0>)]

7

<galactic.algebras.concept.core.Extent object at 0x72ef32f86000>

[Item(key='Eagle', value=<list object at 0x72ef32f648c0>),
 Item(key='Fox', value=<list object at 0x72ef32f64900>),
 Item(key='Dog', value=<list object at 0x72ef32f64940>),
 Item(key='Wolf', value=<list object at 0x72ef32f64980>)]

4

<galactic.algebras.concept.core.Extent object at 0x72ef32f862c0>

[Item(key='Dove', value=<list object at 0x72ef32f4c700>),
 Item(key='Hen', value=<list object at 0x72ef32f647c0>),
 Item(key='Duck', value=<list object at 0x72ef32f64500>),
 Item(key='Goose', value=<list object at 0x72ef32f64800>),
 Item(key='Owl', value=<list object at 0x72ef32f64840>),
 Item(key='Hawk', value=<list object at 0x72ef32f64880>),
 Item(key='Eagle', value=<list object at 0x72ef32f648c0>),
 Item(key='Fox', value=<list object at 0x72ef32f64900>),
 Item(key='Dog', value=<list object at 0x72ef32f64940>),
 Item(key='Wolf', value=<list object at 0x72ef32f64980>),
 Item(key='Cat', value=<list object at 0x72ef32f649c0>),
 Item(key='Tiger', value=<list object at 0x72ef32f64a00>),
 Item(key='Lion', value=<list object at 0x72ef32f64a40>),
 Item(key='Horse', value=<list object at 0x72ef32f64a80>),
 Item(key='Zebra', value=<list object at 0x72ef32f64ac0>),
 Item(key='Cow', value=<list object at 0x72ef32f64b00>)]

16

<galactic.algebras.concept.core.Extent object at 0x72ef32f86400>

[Item(key='Eagle', value=<list object at 0x72ef32f648c0>)]

1

intent_1 = Intent(
    connection,
    [context.domain.item(key='Zebra')],
)
intent_2 = Intent(
    connection,
    [context.domain.item(key='Wolf')],
)
join = intent_1 | intent_2
meet = intent_1 & intent_2
display(intent_1, list(intent_1), len(intent_1))
display(intent_2, list(intent_2), len(intent_2))
display(join, list(join), len(join))
display(meet, list(meet), len(meet))

<galactic.algebras.concept.core.Intent object at 0x72ef4c2f2100>

[<function big at 0x72ef32f559e0>,
 <function fourlegs at 0x72ef32f56160>,
 <function hair at 0x72ef32f56520>,
 <function run at 0x72ef32f57380>,
 <function mane at 0x72ef32f57b00>,
 <function hooves at 0x72ef32f57ec0>]

6

<galactic.algebras.concept.core.Intent object at 0x72ef32f86a80>

[<function medium at 0x72ef32f558a0>,
 <function fourlegs at 0x72ef32f56160>,
 <function hair at 0x72ef32f56520>,
 <function run at 0x72ef32f57380>,
 <function hunt at 0x72ef32f57740>,
 <function mane at 0x72ef32f57b00>]

6

<galactic.algebras.concept.core.Intent object at 0x72ef519b4d00>

[<function small at 0x72ef32f540e0>,
 <function medium at 0x72ef32f558a0>,
 <function big at 0x72ef32f559e0>,
 <function twolegs at 0x72ef32f55da0>,
 <function fourlegs at 0x72ef32f56160>,
 <function hair at 0x72ef32f56520>,
 <function feathers at 0x72ef32f55580>,
 <function fly at 0x72ef32f56c00>,
 <function swim at 0x72ef32f56fc0>,
 <function run at 0x72ef32f57380>,
 <function hunt at 0x72ef32f57740>,
 <function mane at 0x72ef32f57b00>,
 <function hooves at 0x72ef32f57ec0>]

13

<galactic.algebras.concept.core.Intent object at 0x72ef4c2c7300>

[<function fourlegs at 0x72ef32f56160>,
 <function hair at 0x72ef32f56520>,
 <function run at 0x72ef32f57380>,
 <function mane at 0x72ef32f57b00>]

4

Iterable given in arguments to constructors can be any iterable including instance of the other closed set class (in that case, the operation is optimized since intents and extents are represented by bitmaps).

extent_from_intent = Extent(
    connection,
    intent_1,
)
display(extent_from_intent, list(extent_from_intent), len(extent_from_intent))
intent_from_extent = Intent(
    connection,
    extent_1,
)
display(intent_from_extent, list(intent_from_extent), len(intent_from_extent))

<galactic.algebras.concept.core.Extent object at 0x72ef32f86f40>

[Item(key='Horse', value=<list object at 0x72ef32f64a80>),
 Item(key='Zebra', value=<list object at 0x72ef32f64ac0>)]

2

<galactic.algebras.concept.core.Intent object at 0x72ef32f85340>

[<function twolegs at 0x72ef32f55da0>, <function feathers at 0x72ef32f55580>]

2