📘 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 0x7859a46b2980>

(<galactic.algebras.concept.core.AttrPolarity object at 0x7859a46c9b40>,
 <galactic.algebras.concept.core.ItemPolarity object at 0x7859a46c9b70>)

(compose(<galactic.algebras.concept.core.ItemPolarity object at 0x7859a46c9b70>, <galactic.algebras.concept.core.AttrPolarity object at 0x7859a46c9b40>),
 compose(<galactic.algebras.concept.core.AttrPolarity object at 0x7859a46c9b40>, <galactic.algebras.concept.core.ItemPolarity object at 0x7859a46c9b70>))

<galactic.algebras.concept.core.Context object at 0x7859adef45e0>

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 0x7859a46c9b70>, <galactic.algebras.concept.core.AttrPolarity object at 0x7859a46c9b40>)

compose(<galactic.algebras.concept.core.AttrPolarity object at 0x7859a46c9b40>, <galactic.algebras.concept.core.ItemPolarity object at 0x7859a46c9b70>)

<galactic.algebras.concept.core.Extent object at 0x7859a46cdf40>

[Item(key='Fox', value=<list object at 0x7859a46b3000>),
 Item(key='Dog', value=<list object at 0x7859a46b3040>),
 Item(key='Wolf', value=<list object at 0x7859a46b3080>),
 Item(key='Cat', value=<list object at 0x7859a46b30c0>),
 Item(key='Tiger', value=<list object at 0x7859a46b3100>),
 Item(key='Lion', value=<list object at 0x7859a46b3140>),
 Item(key='Horse', value=<list object at 0x7859a46b3180>),
 Item(key='Zebra', value=<list object at 0x7859a46b31c0>)]

8

<galactic.algebras.concept.core.Intent object at 0x7859a46e1140>

[<function twolegs at 0x7859a46c4ae0>, <function feathers at 0x7859a46af100>]

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 0x7859a46e2380>

[Item(key='Dove', value=<list object at 0x7859a46b2b40>),
 Item(key='Hen', value=<list object at 0x7859a46b2e80>),
 Item(key='Duck', value=<list object at 0x7859a46b2ec0>),
 Item(key='Goose', value=<list object at 0x7859a46b2f00>),
 Item(key='Owl', value=<list object at 0x7859a46b2f40>),
 Item(key='Hawk', value=<list object at 0x7859a46b2f80>),
 Item(key='Eagle', value=<list object at 0x7859a46b2fc0>)]

7

<galactic.algebras.concept.core.Extent object at 0x7859a46e2200>

[Item(key='Eagle', value=<list object at 0x7859a46b2fc0>),
 Item(key='Fox', value=<list object at 0x7859a46b3000>),
 Item(key='Dog', value=<list object at 0x7859a46b3040>),
 Item(key='Wolf', value=<list object at 0x7859a46b3080>)]

4

<galactic.algebras.concept.core.Extent object at 0x7859ac210840>

[Item(key='Dove', value=<list object at 0x7859a46b2b40>),
 Item(key='Hen', value=<list object at 0x7859a46b2e80>),
 Item(key='Duck', value=<list object at 0x7859a46b2ec0>),
 Item(key='Goose', value=<list object at 0x7859a46b2f00>),
 Item(key='Owl', value=<list object at 0x7859a46b2f40>),
 Item(key='Hawk', value=<list object at 0x7859a46b2f80>),
 Item(key='Eagle', value=<list object at 0x7859a46b2fc0>),
 Item(key='Fox', value=<list object at 0x7859a46b3000>),
 Item(key='Dog', value=<list object at 0x7859a46b3040>),
 Item(key='Wolf', value=<list object at 0x7859a46b3080>),
 Item(key='Cat', value=<list object at 0x7859a46b30c0>),
 Item(key='Tiger', value=<list object at 0x7859a46b3100>),
 Item(key='Lion', value=<list object at 0x7859a46b3140>),
 Item(key='Horse', value=<list object at 0x7859a46b3180>),
 Item(key='Zebra', value=<list object at 0x7859a46b31c0>),
 Item(key='Cow', value=<list object at 0x7859a46b3200>)]

16

<galactic.algebras.concept.core.Extent object at 0x7859a46e2600>

[Item(key='Eagle', value=<list object at 0x7859a46b2fc0>)]

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 0x7859ac045580>

[<function big at 0x7859a46c4720>,
 <function fourlegs at 0x7859a46c4ea0>,
 <function hair at 0x7859a46c5260>,
 <function run at 0x7859a46c60c0>,
 <function mane at 0x7859a46c6840>,
 <function hooves at 0x7859a46c6c00>]

6

<galactic.algebras.concept.core.Intent object at 0x7859a46e0380>

[<function medium at 0x7859a46c45e0>,
 <function fourlegs at 0x7859a46c4ea0>,
 <function hair at 0x7859a46c5260>,
 <function run at 0x7859a46c60c0>,
 <function hunt at 0x7859a46c6480>,
 <function mane at 0x7859a46c6840>]

6

<galactic.algebras.concept.core.Intent object at 0x7859a46cd240>

[<function small at 0x7859a46831a0>,
 <function medium at 0x7859a46c45e0>,
 <function big at 0x7859a46c4720>,
 <function twolegs at 0x7859a46c4ae0>,
 <function fourlegs at 0x7859a46c4ea0>,
 <function hair at 0x7859a46c5260>,
 <function feathers at 0x7859a46af100>,
 <function fly at 0x7859a46c5940>,
 <function swim at 0x7859a46c5d00>,
 <function run at 0x7859a46c60c0>,
 <function hunt at 0x7859a46c6480>,
 <function mane at 0x7859a46c6840>,
 <function hooves at 0x7859a46c6c00>]

13

<galactic.algebras.concept.core.Intent object at 0x7859a46e11c0>

[<function fourlegs at 0x7859a46c4ea0>,
 <function hair at 0x7859a46c5260>,
 <function run at 0x7859a46c60c0>,
 <function mane at 0x7859a46c6840>]

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 0x7859ac045800>

[Item(key='Horse', value=<list object at 0x7859a46b3180>),
 Item(key='Zebra', value=<list object at 0x7859a46b31c0>)]

2

<galactic.algebras.concept.core.Intent object at 0x7859ac045680>

[<function twolegs at 0x7859a46c4ae0>, <function feathers at 0x7859a46af100>]

2