📘 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 0x7fc432ff83c0>

(<galactic.algebras.concept.core.AttrPolarity object at 0x7fc433022f80>,
 <galactic.algebras.concept.core.ItemPolarity object at 0x7fc433022fb0>)

(compose(<galactic.algebras.concept.core.ItemPolarity object at 0x7fc433022fb0>, <galactic.algebras.concept.core.AttrPolarity object at 0x7fc433022f80>),
 compose(<galactic.algebras.concept.core.AttrPolarity object at 0x7fc433022f80>, <galactic.algebras.concept.core.ItemPolarity object at 0x7fc433022fb0>))

<galactic.algebras.concept.core.Context object at 0x7fc4338b5670>

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 0x7fc433022fb0>, <galactic.algebras.concept.core.AttrPolarity object at 0x7fc433022f80>)

compose(<galactic.algebras.concept.core.AttrPolarity object at 0x7fc433022f80>, <galactic.algebras.concept.core.ItemPolarity object at 0x7fc433022fb0>)

<galactic.algebras.concept.core.Extent object at 0x7fc4338aea40>

[Item(key='Fox', value=<list object at 0x7fc433028c40>),
 Item(key='Dog', value=<list object at 0x7fc433028c80>),
 Item(key='Wolf', value=<list object at 0x7fc433028cc0>),
 Item(key='Cat', value=<list object at 0x7fc433028d00>),
 Item(key='Tiger', value=<list object at 0x7fc433028d40>),
 Item(key='Lion', value=<list object at 0x7fc433028d80>),
 Item(key='Horse', value=<list object at 0x7fc433028dc0>),
 Item(key='Zebra', value=<list object at 0x7fc433028e00>)]

8

<galactic.algebras.concept.core.Intent object at 0x7fc433038c40>

[<function twolegs at 0x7fc43302c9a0>, <function feathers at 0x7fc433012fc0>]

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=connection,
    attrs=[context.co_domain.attr(name='feathers')],
)
extent_2 = Extent(
    connection=connection,
    attrs=[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 0x7fc43304a740>

[Item(key='Dove', value=<list object at 0x7fc4330286c0>),
 Item(key='Hen', value=<list object at 0x7fc433028b00>),
 Item(key='Duck', value=<list object at 0x7fc433028800>),
 Item(key='Goose', value=<list object at 0x7fc433028b40>),
 Item(key='Owl', value=<list object at 0x7fc433028b80>),
 Item(key='Hawk', value=<list object at 0x7fc433028bc0>),
 Item(key='Eagle', value=<list object at 0x7fc433028c00>)]

7

<galactic.algebras.concept.core.Extent object at 0x7fc43304a300>

[Item(key='Eagle', value=<list object at 0x7fc433028c00>),
 Item(key='Fox', value=<list object at 0x7fc433028c40>),
 Item(key='Dog', value=<list object at 0x7fc433028c80>),
 Item(key='Wolf', value=<list object at 0x7fc433028cc0>)]

4

<galactic.algebras.concept.core.Extent object at 0x7fc433852140>

[Item(key='Dove', value=<list object at 0x7fc4330286c0>),
 Item(key='Hen', value=<list object at 0x7fc433028b00>),
 Item(key='Duck', value=<list object at 0x7fc433028800>),
 Item(key='Goose', value=<list object at 0x7fc433028b40>),
 Item(key='Owl', value=<list object at 0x7fc433028b80>),
 Item(key='Hawk', value=<list object at 0x7fc433028bc0>),
 Item(key='Eagle', value=<list object at 0x7fc433028c00>),
 Item(key='Fox', value=<list object at 0x7fc433028c40>),
 Item(key='Dog', value=<list object at 0x7fc433028c80>),
 Item(key='Wolf', value=<list object at 0x7fc433028cc0>),
 Item(key='Cat', value=<list object at 0x7fc433028d00>),
 Item(key='Tiger', value=<list object at 0x7fc433028d40>),
 Item(key='Lion', value=<list object at 0x7fc433028d80>),
 Item(key='Horse', value=<list object at 0x7fc433028dc0>),
 Item(key='Zebra', value=<list object at 0x7fc433028e00>),
 Item(key='Cow', value=<list object at 0x7fc433028e40>)]

16

<galactic.algebras.concept.core.Extent object at 0x7fc43304a240>

[Item(key='Eagle', value=<list object at 0x7fc433028c00>)]

1

intent_1 = Intent(
    connection=connection,
    items=[context.domain.item(key='Zebra')],
)
intent_2 = Intent(
    connection=connection,
    items=[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 0x7fc433049980>

[<function big at 0x7fc43302c5e0>,
 <function fourlegs at 0x7fc43302cd60>,
 <function hair at 0x7fc43302d120>,
 <function run at 0x7fc43302df80>,
 <function mane at 0x7fc43302e700>,
 <function hooves at 0x7fc43302eac0>]

6

<galactic.algebras.concept.core.Intent object at 0x7fc433049900>

[<function medium at 0x7fc43302c4a0>,
 <function fourlegs at 0x7fc43302cd60>,
 <function hair at 0x7fc43302d120>,
 <function run at 0x7fc43302df80>,
 <function hunt at 0x7fc43302e340>,
 <function mane at 0x7fc43302e700>]

6

<galactic.algebras.concept.core.Intent object at 0x7fc43329a540>

[<function small at 0x7fc4338dcae0>,
 <function medium at 0x7fc43302c4a0>,
 <function big at 0x7fc43302c5e0>,
 <function twolegs at 0x7fc43302c9a0>,
 <function fourlegs at 0x7fc43302cd60>,
 <function hair at 0x7fc43302d120>,
 <function feathers at 0x7fc433012fc0>,
 <function fly at 0x7fc43302d800>,
 <function swim at 0x7fc43302dbc0>,
 <function run at 0x7fc43302df80>,
 <function hunt at 0x7fc43302e340>,
 <function mane at 0x7fc43302e700>,
 <function hooves at 0x7fc43302eac0>]

13

<galactic.algebras.concept.core.Intent object at 0x7fc433049f40>

[<function fourlegs at 0x7fc43302cd60>,
 <function hair at 0x7fc43302d120>,
 <function run at 0x7fc43302df80>,
 <function mane at 0x7fc43302e700>]

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

<galactic.algebras.concept.core.Extent object at 0x7fc43303b1c0>

[Item(key='Horse', value=<list object at 0x7fc433028dc0>),
 Item(key='Zebra', value=<list object at 0x7fc433028e00>)]

2

<galactic.algebras.concept.core.Intent object at 0x7fc43303bb40>

[<function twolegs at 0x7fc43302c9a0>, <function feathers at 0x7fc433012fc0>]

2