📘 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 0x7485f95ffe40>

(<galactic.algebras.concept.core.AttrPolarity object at 0x7485f9640c40>,
 <galactic.algebras.concept.core.ItemPolarity object at 0x7485f8cea260>)

(compose(<galactic.algebras.concept.core.ItemPolarity object at 0x7485f8cea260>, <galactic.algebras.concept.core.AttrPolarity object at 0x7485f9640c40>),
 compose(<galactic.algebras.concept.core.AttrPolarity object at 0x7485f9640c40>, <galactic.algebras.concept.core.ItemPolarity object at 0x7485f8cea260>))

<galactic.algebras.concept.core.Context object at 0x74860ac548b0>

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 0x7485f8cea260>, <galactic.algebras.concept.core.AttrPolarity object at 0x7485f9640c40>)

compose(<galactic.algebras.concept.core.AttrPolarity object at 0x7485f9640c40>, <galactic.algebras.concept.core.ItemPolarity object at 0x7485f8cea260>)

<galactic.algebras.concept.core.Extent object at 0x7485f8c8dfc0>

[Item(key='Fox', value=<list object at 0x7485f8cd7500>),
 Item(key='Dog', value=<list object at 0x7485f8cd7540>),
 Item(key='Wolf', value=<list object at 0x7485f8cd7580>),
 Item(key='Cat', value=<list object at 0x7485f8cd75c0>),
 Item(key='Tiger', value=<list object at 0x7485f8cd7600>),
 Item(key='Lion', value=<list object at 0x7485f8cd7640>),
 Item(key='Horse', value=<list object at 0x7485f8cd7680>),
 Item(key='Zebra', value=<list object at 0x7485f8cd76c0>)]

8

<galactic.algebras.concept.core.Intent object at 0x7485f8cfc640>

[<function twolegs at 0x7485f8ce4fe0>, <function feathers at 0x7485f8cd3600>]

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 0x7485f8cfd040>

[Item(key='Dove', value=<list object at 0x7485f8cd7040>),
 Item(key='Hen', value=<list object at 0x7485f8cd7380>),
 Item(key='Duck', value=<list object at 0x7485f8cd73c0>),
 Item(key='Goose', value=<list object at 0x7485f8cd7400>),
 Item(key='Owl', value=<list object at 0x7485f8cd7440>),
 Item(key='Hawk', value=<list object at 0x7485f8cd7480>),
 Item(key='Eagle', value=<list object at 0x7485f8cd74c0>)]

7

<galactic.algebras.concept.core.Extent object at 0x7485f8cfd1c0>

[Item(key='Eagle', value=<list object at 0x7485f8cd74c0>),
 Item(key='Fox', value=<list object at 0x7485f8cd7500>),
 Item(key='Dog', value=<list object at 0x7485f8cd7540>),
 Item(key='Wolf', value=<list object at 0x7485f8cd7580>)]

4

<galactic.algebras.concept.core.Extent object at 0x7485f9621e40>

[Item(key='Dove', value=<list object at 0x7485f8cd7040>),
 Item(key='Hen', value=<list object at 0x7485f8cd7380>),
 Item(key='Duck', value=<list object at 0x7485f8cd73c0>),
 Item(key='Goose', value=<list object at 0x7485f8cd7400>),
 Item(key='Owl', value=<list object at 0x7485f8cd7440>),
 Item(key='Hawk', value=<list object at 0x7485f8cd7480>),
 Item(key='Eagle', value=<list object at 0x7485f8cd74c0>),
 Item(key='Fox', value=<list object at 0x7485f8cd7500>),
 Item(key='Dog', value=<list object at 0x7485f8cd7540>),
 Item(key='Wolf', value=<list object at 0x7485f8cd7580>),
 Item(key='Cat', value=<list object at 0x7485f8cd75c0>),
 Item(key='Tiger', value=<list object at 0x7485f8cd7600>),
 Item(key='Lion', value=<list object at 0x7485f8cd7640>),
 Item(key='Horse', value=<list object at 0x7485f8cd7680>),
 Item(key='Zebra', value=<list object at 0x7485f8cd76c0>),
 Item(key='Cow', value=<list object at 0x7485f8cd7700>)]

16

<galactic.algebras.concept.core.Extent object at 0x7485f8cfd500>

[Item(key='Eagle', value=<list object at 0x7485f8cd74c0>)]

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 0x7485f8cfd700>

[<function big at 0x7485f8ce4c20>,
 <function fourlegs at 0x7485f8ce53a0>,
 <function hair at 0x7485f8ce5760>,
 <function run at 0x7485f8ce65c0>,
 <function mane at 0x7485f8ce6d40>,
 <function hooves at 0x7485f8ce7100>]

6

<galactic.algebras.concept.core.Intent object at 0x7485f8cfc480>

[<function medium at 0x7485f8ce4ae0>,
 <function fourlegs at 0x7485f8ce53a0>,
 <function hair at 0x7485f8ce5760>,
 <function run at 0x7485f8ce65c0>,
 <function hunt at 0x7485f8ce6980>,
 <function mane at 0x7485f8ce6d40>]

6

<galactic.algebras.concept.core.Intent object at 0x7485f8cfcbc0>

[<function small at 0x7485f8c9f6a0>,
 <function medium at 0x7485f8ce4ae0>,
 <function big at 0x7485f8ce4c20>,
 <function twolegs at 0x7485f8ce4fe0>,
 <function fourlegs at 0x7485f8ce53a0>,
 <function hair at 0x7485f8ce5760>,
 <function feathers at 0x7485f8cd3600>,
 <function fly at 0x7485f8ce5e40>,
 <function swim at 0x7485f8ce6200>,
 <function run at 0x7485f8ce65c0>,
 <function hunt at 0x7485f8ce6980>,
 <function mane at 0x7485f8ce6d40>,
 <function hooves at 0x7485f8ce7100>]

13

<galactic.algebras.concept.core.Intent object at 0x7485f8cfdfc0>

[<function fourlegs at 0x7485f8ce53a0>,
 <function hair at 0x7485f8ce5760>,
 <function run at 0x7485f8ce65c0>,
 <function mane at 0x7485f8ce6d40>]

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 0x7485f8cff940>

[Item(key='Horse', value=<list object at 0x7485f8cd7680>),
 Item(key='Zebra', value=<list object at 0x7485f8cd76c0>)]

2

<galactic.algebras.concept.core.Intent object at 0x7485f8cff300>

[<function twolegs at 0x7485f8ce4fe0>, <function feathers at 0x7485f8cd3600>]

2