Lattice

Elements

Lattice are a particular case of poset. Their elements have the following properties:

For all $x, y$ of a poset, there exists a unique infimum and a unique supremum:

Integer example

[1]:

from galactic.examples.arithmetic import Integer

[2]:

Integer(24) & Integer(18)

[2]:

$6=2^13^1$

[3]:

Integer(24) | Integer(18)

[3]:

$72=2^33^2$

Color example

[4]:

from galactic.examples.color import Color

[5]:

color1 = Color(red=0.8, green=0.5, blue=0.7)
color1

[5]:

../../../_images/notebooks_algebras_lattice_notebook_6_0.svg

[6]:

color2 = Color(red=0.5, green=1.0, blue=0.9)
color2

[6]:

../../../_images/notebooks_algebras_lattice_notebook_7_0.svg

[7]:

color1 & color2

[7]:

../../../_images/notebooks_algebras_lattice_notebook_8_0.svg

[8]:

color1 | color2

[8]:

../../../_images/notebooks_algebras_lattice_notebook_9_0.svg

Collections

All operations on poset are available and some extras features:

get the minimum and maximum elements from a lattice;
get the atoms and the co-atoms elements from a lattice;
get the meet-irreducible and the join-irreducible elements from a lattice;
get the meet-irreducible generators and the join-irreducible generators of an element of the lattice.

Integer example

[9]:

from galactic.algebras.lattice import (
    Lattice,
    ReducedContextDiagram
)
from galactic.algebras.poset import HasseDiagram
from galactic.examples.arithmetic import IntegerRenderer

[10]:

integers = [
    Integer(2),
    Integer(3),
    Integer(5),
    Integer(7),
    Integer(9)
]
lattice = Lattice[Integer](domain=integers)
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(
        meet_semi_lattice=True,
        join_semi_lattice=True,
    )
)

[10]:

../../../_images/notebooks_algebras_lattice_notebook_13_0.png

[11]:

lattice.minimum

[11]:

$1$

[12]:

lattice.maximum

[12]:

$630=2^13^25^17^1$

[13]:

display(*lattice.atoms)

$2=2^1$

$3=3^1$

$5=5^1$

$7=7^1$

[14]:

display(*lattice.co_atoms)

$210=2^13^15^17^1$

$126=2^13^27^1$

$315=3^25^17^1$

$90=2^13^25^1$

[15]:

display(*lattice.meet_irreducible)

$210=2^13^15^17^1$

$126=2^13^27^1$

$315=3^25^17^1$

$90=2^13^25^1$

$70=2^15^17^1$

[16]:

display(*lattice.join_irreducible)

$2=2^1$

$3=3^1$

$5=5^1$

$7=7^1$

$9=3^2$

[17]:

display(*lattice.smallest_meet_irreducible(Integer(30)))

$210=2^13^15^17^1$

$90=2^13^25^1$

[18]:

display(*lattice.greatest_join_irreducible(Integer(30)))

$2=2^1$

$3=3^1$

$5=5^1$

Color example

[19]:

from galactic.examples.color import Color, ColorRenderer
from galactic.algebras.lattice import Lattice

[20]:

color3 = Color(red=1.0, green=0.7, blue=0.5)
color3

[20]:

../../../_images/notebooks_algebras_lattice_notebook_24_0.svg

[21]:

color4 = Color(red=0.6, green=1.0, blue=0.1)
color4

[21]:

../../../_images/notebooks_algebras_lattice_notebook_25_0.svg

[22]:

color5 = Color(red=0.4, green=0.6, blue=0.7)
color5

[22]:

../../../_images/notebooks_algebras_lattice_notebook_26_0.svg

[23]:

lattice = Lattice[Color](
    domain = [
        color3,
        color4,
        color5,
    ]
)

[24]:

HasseDiagram[Color](lattice, domain_renderer=ColorRenderer())

[24]:

../../../_images/notebooks_algebras_lattice_notebook_28_0.png

[25]:

lattice.extend([
    color1
])

[26]:

HasseDiagram[Color](lattice, domain_renderer=ColorRenderer())

[26]:

../../../_images/notebooks_algebras_lattice_notebook_30_0.png

[27]:

ReducedContextDiagram(
    lattice,
    domain_renderer=ColorRenderer(),
    co_domain_renderer=ColorRenderer(),
)

[27]:

../../../_images/notebooks_algebras_lattice_notebook_31_0.png

Context

There exists a bijection between a lattice and its minimal binary table called its equivalent context:

the rows are composed by the meet-irreducible
the columns are composed by the join-irreducible
the boolean value for row $i$ and column $j$ is True if $i\geq j$

Integer example

[28]:

integers = [
    Integer(2),
    Integer(3),
    Integer(5),
    Integer(7),
    Integer(9)
]
lattice = Lattice[Integer](domain=integers)
ReducedContextDiagram(
    lattice,
    domain_renderer=IntegerRenderer(join_irreducible=True),
    co_domain_renderer=IntegerRenderer(meet_irreducible=True),
)

[28]:

../../../_images/notebooks_algebras_lattice_notebook_33_0.png

[29]:

from galactic.algebras.relational import BinaryTable
BinaryTable(
    lattice.reduced_context,
    domain_renderer=IntegerRenderer(join_irreducible=True),
    co_domain_renderer=IntegerRenderer(meet_irreducible=True)
)

[29]:

	210	126	315	90	70
$2=2^1$	$\checkmark$	$\checkmark$		$\checkmark$	$\checkmark$
$3=3^1$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$
$5=5^1$	$\checkmark$		$\checkmark$	$\checkmark$	$\checkmark$
$7=7^1$	$\checkmark$	$\checkmark$	$\checkmark$		$\checkmark$
$9=3^2$		$\checkmark$	$\checkmark$	$\checkmark$

	210	126	315	90	70
\(2=2^1\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)
\(3=3^1\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(5=5^1\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(7=7^1\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)
\(9=3^2\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)