Finite Lattices

Elements

Finite lattices are a particular case of posets. Their elements have the following properties:

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

PrimeFactors example

[1]:

from galactic.algebras.examples.arithmetic import PrimeFactors

[2]:

PrimeFactors(24) & PrimeFactors(18)

[2]:

$6=2·3$

[3]:

PrimeFactors(24) | PrimeFactors(18)

[3]:

$72=2^3·3^2$

Color example

[4]:

from galactic.algebras.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.

PrimeFactors example

[9]:

from galactic.algebras.examples.arithmetic.renderer import PrimeFactorsRenderer
from galactic.algebras.lattice import ExtensibleFiniteLattice
from galactic.algebras.lattice.renderer import ReducedContextDiagram
from galactic.algebras.poset.renderer import HasseDiagram

[10]:

integers = [
    PrimeFactors(2),
    PrimeFactors(3),
    PrimeFactors(5),
    PrimeFactors(7),
    PrimeFactors(9),
]
lattice = ExtensibleFiniteLattice[PrimeFactors](elements=integers)
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        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]:

$630=2·3^2·5·7$

[13]:

display(*lattice.atoms)

$3=3$

$7=7$

$2=2$

$5=5$

[14]:

display(*lattice.co_atoms)

$315=3^2·5·7$

$126=2·3^2·7$

$90=2·3^2·5$

$210=2·3·5·7$

[15]:

display(*lattice.meet_irreducible)

$70=2·5·7$

$210=2·3·5·7$

$90=2·3^2·5$

$126=2·3^2·7$

$315=3^2·5·7$

[16]:

display(*lattice.join_irreducible)

$3=3$

$9=3^2$

$7=7$

$2=2$

$5=5$

[17]:

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

$210=2·3·5·7$

$90=2·3^2·5$

[18]:

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

$3=3$

$2=2$

$5=5$

Color example

[19]:

from galactic.algebras.examples.color import Color
from galactic.algebras.examples.color.renderer import ColorRenderer
from galactic.algebras.lattice import ExtensibleFiniteLattice

[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 = ExtensibleFiniteLattice[Color](
    elements=[
        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 $i\geq j$

PrimeFactors example

[28]:

integers = [
    PrimeFactors(2),
    PrimeFactors(3),
    PrimeFactors(5),
    PrimeFactors(7),
    PrimeFactors(9),
]
lattice = ExtensibleFiniteLattice[PrimeFactors](elements=integers)
ReducedContextDiagram(
    lattice,
    domain_renderer=PrimeFactorsRenderer(join_irreducible=True),
    co_domain_renderer=PrimeFactorsRenderer(meet_irreducible=True),
)

[28]:

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

[29]:

from galactic.algebras.relational.renderer import BinaryTable

BinaryTable(
    lattice.reduced_context,
    domain_renderer=PrimeFactorsRenderer(join_irreducible=True),
    co_domain_renderer=PrimeFactorsRenderer(meet_irreducible=True),
)

[29]:

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