📘 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 0x761da9e2e580>

(<galactic.algebras.concept.core.AttrPolarity object at 0x761da94ea380>,
 <galactic.algebras.concept.core.ItemPolarity object at 0x761da94ea3b0>)

(compose(<galactic.algebras.concept.core.ItemPolarity object at 0x761da94ea3b0>, <galactic.algebras.concept.core.AttrPolarity object at 0x761da94ea380>),
 compose(<galactic.algebras.concept.core.AttrPolarity object at 0x761da94ea380>, <galactic.algebras.concept.core.ItemPolarity object at 0x761da94ea3b0>))

<galactic.algebras.concept.core.Context object at 0x761db836e610>

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 0x761da94ea3b0>, <galactic.algebras.concept.core.AttrPolarity object at 0x761da94ea380>)

compose(<galactic.algebras.concept.core.AttrPolarity object at 0x761da94ea380>, <galactic.algebras.concept.core.ItemPolarity object at 0x761da94ea3b0>)

<galactic.algebras.concept.core.Extent object at 0x761da9e3e700>

[Item(key='Fox', value=<list object at 0x761da94d7e40>),
 Item(key='Dog', value=<list object at 0x761da94d7e80>),
 Item(key='Wolf', value=<list object at 0x761da94d7ec0>),
 Item(key='Cat', value=<list object at 0x761da94d7f00>),
 Item(key='Tiger', value=<list object at 0x761da94d7f40>),
 Item(key='Lion', value=<list object at 0x761da94d7f80>),
 Item(key='Horse', value=<list object at 0x761da94d7fc0>),
 Item(key='Zebra', value=<list object at 0x761da94f0040>)]

8

<galactic.algebras.concept.core.Intent object at 0x761da94f2ec0>

[<function twolegs at 0x761da94e4fe0>, <function feathers at 0x761da94d3600>]

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 0x761da9505640>

[Item(key='Dove', value=<list object at 0x761da94d7980>),
 Item(key='Hen', value=<list object at 0x761da94d7cc0>),
 Item(key='Duck', value=<list object at 0x761da94d7d00>),
 Item(key='Goose', value=<list object at 0x761da94d7d40>),
 Item(key='Owl', value=<list object at 0x761da94d7d80>),
 Item(key='Hawk', value=<list object at 0x761da94d7dc0>),
 Item(key='Eagle', value=<list object at 0x761da94d7e00>)]

7

<galactic.algebras.concept.core.Extent object at 0x761da9505740>

[Item(key='Eagle', value=<list object at 0x761da94d7e00>),
 Item(key='Fox', value=<list object at 0x761da94d7e40>),
 Item(key='Dog', value=<list object at 0x761da94d7e80>),
 Item(key='Wolf', value=<list object at 0x761da94d7ec0>)]

4

<galactic.algebras.concept.core.Extent object at 0x761da9505ac0>

[Item(key='Dove', value=<list object at 0x761da94d7980>),
 Item(key='Hen', value=<list object at 0x761da94d7cc0>),
 Item(key='Duck', value=<list object at 0x761da94d7d00>),
 Item(key='Goose', value=<list object at 0x761da94d7d40>),
 Item(key='Owl', value=<list object at 0x761da94d7d80>),
 Item(key='Hawk', value=<list object at 0x761da94d7dc0>),
 Item(key='Eagle', value=<list object at 0x761da94d7e00>),
 Item(key='Fox', value=<list object at 0x761da94d7e40>),
 Item(key='Dog', value=<list object at 0x761da94d7e80>),
 Item(key='Wolf', value=<list object at 0x761da94d7ec0>),
 Item(key='Cat', value=<list object at 0x761da94d7f00>),
 Item(key='Tiger', value=<list object at 0x761da94d7f40>),
 Item(key='Lion', value=<list object at 0x761da94d7f80>),
 Item(key='Horse', value=<list object at 0x761da94d7fc0>),
 Item(key='Zebra', value=<list object at 0x761da94f0040>),
 Item(key='Cow', value=<list object at 0x761da94f0080>)]

16

<galactic.algebras.concept.core.Extent object at 0x761da9e3cd40>

[Item(key='Eagle', value=<list object at 0x761da94d7e00>)]

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 0x761da9e16bc0>

[<function big at 0x761da94e4c20>,
 <function fourlegs at 0x761da94e53a0>,
 <function hair at 0x761da94e5760>,
 <function run at 0x761da94e65c0>,
 <function mane at 0x761da94e6d40>,
 <function hooves at 0x761da94e7100>]

6

<galactic.algebras.concept.core.Intent object at 0x761da9504d00>

[<function medium at 0x761da94e4ae0>,
 <function fourlegs at 0x761da94e53a0>,
 <function hair at 0x761da94e5760>,
 <function run at 0x761da94e65c0>,
 <function hunt at 0x761da94e6980>,
 <function mane at 0x761da94e6d40>]

6

<galactic.algebras.concept.core.Intent object at 0x761dbbc69e40>

[<function small at 0x761da949f6a0>,
 <function medium at 0x761da94e4ae0>,
 <function big at 0x761da94e4c20>,
 <function twolegs at 0x761da94e4fe0>,
 <function fourlegs at 0x761da94e53a0>,
 <function hair at 0x761da94e5760>,
 <function feathers at 0x761da94d3600>,
 <function fly at 0x761da94e5e40>,
 <function swim at 0x761da94e6200>,
 <function run at 0x761da94e65c0>,
 <function hunt at 0x761da94e6980>,
 <function mane at 0x761da94e6d40>,
 <function hooves at 0x761da94e7100>]

13

<galactic.algebras.concept.core.Intent object at 0x761da94d4d80>

[<function fourlegs at 0x761da94e53a0>,
 <function hair at 0x761da94e5760>,
 <function run at 0x761da94e65c0>,
 <function mane at 0x761da94e6d40>]

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 0x761da9507f40>

[Item(key='Horse', value=<list object at 0x761da94d7fc0>),
 Item(key='Zebra', value=<list object at 0x761da94f0040>)]

2

<galactic.algebras.concept.core.Intent object at 0x761da9506440>

[<function twolegs at 0x761da94e4fe0>, <function feathers at 0x761da94d3600>]

2