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

from galactic.examples.arithmetic.algebras import Integer

Integer(24) & Integer(18)

$6=2^13^1$

Integer(24) | Integer(18)

$72=2^33^2$

Color example

from galactic.examples.color.algebras import Color

Color(red=0.8, green=0.5, blue=0.7) \
    & Color(red=0.5, green=1.0, blue=0.9)

Color(red=0.8, green=0.5, blue=0.7) \
    | Color(red=0.5, green=1.0, blue=0.9)

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

from galactic.algebras.lattice.collections import BasicLattice
from galactic.algebras.poset.collections import HasseDiagram

integers = [
    Integer(2),
    Integer(3),
    Integer(5),
    Integer(7),
    Integer(9)
]
lattice = BasicLattice(integers)
HasseDiagram(lattice)

display(*lattice)

$2=2^1$

$3=3^1$

$5=5^1$

$6=2^13^1$

$7=7^1$

$9=3^2$

$10=2^15^1$

$42=2^13^17^1$

$45=3^25^1$

$14=2^17^1$

$15=3^15^1$

$18=2^13^2$

$21=3^17^1$

$30=2^13^15^1$

$35=5^17^1$

$315=3^25^17^1$

$63=3^27^1$

$70=2^15^17^1$

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

$90=2^13^25^1$

$105=3^15^17^1$

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

$126=2^13^27^1$

$1$

lattice.minimum()

$1$

lattice.maximum()

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

display(*lattice.atoms())

$2=2^1$

$3=3^1$

$5=5^1$

$7=7^1$

display(*lattice.co_atoms())

$315=3^25^17^1$

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

$90=2^13^25^1$

$126=2^13^27^1$

display(*lattice.meet_irreducible())

$315=3^25^17^1$

$70=2^15^17^1$

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

$90=2^13^25^1$

$126=2^13^27^1$

display(*lattice.join_irreducible())

$2=2^1$

$3=3^1$

$5=5^1$

$7=7^1$

$9=3^2$

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

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

$90=2^13^25^1$

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

$2=2^1$

$3=3^1$

$5=5^1$

Color example

from galactic.examples.color.algebras import Color
from galactic.algebras.lattice.collections import CompactLattice

lattice = CompactLattice([
    Color(red=1.0, green=0.7, blue=0.5),
    Color(red=0.6, green=1.0, blue=0.1),
    Color(red=0.4, green=0.6, blue=0.7),
    Color(red=1.0, green=0.4, blue=0.3),
])

HasseDiagram(lattice)

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

from galactic.algebras.lattice.contexts import ReducedContext
integers = [
    Integer(2),
    Integer(3),
    Integer(5),
    Integer(7),
    Integer(9)
]
lattice = CompactLattice(integers)
context = ReducedContext(lattice)
list(context)

[(Integer(2), Integer(70)),
 (Integer(2), Integer(90)),
 (Integer(2), Integer(126)),
 (Integer(2), Integer(210)),
 (Integer(3), Integer(90)),
 (Integer(3), Integer(126)),
 (Integer(3), Integer(210)),
 (Integer(3), Integer(315)),
 (Integer(5), Integer(70)),
 (Integer(5), Integer(90)),
 (Integer(5), Integer(210)),
 (Integer(5), Integer(315)),
 (Integer(7), Integer(70)),
 (Integer(7), Integer(126)),
 (Integer(7), Integer(210)),
 (Integer(7), Integer(315)),
 (Integer(9), Integer(90)),
 (Integer(9), Integer(126)),
 (Integer(9), Integer(315))]

context

	\(70=2^15^17^1\)	\(90=2^13^25^1\)	\(126=2^13^27^1\)	\(210=2^13^15^17^1\)	\(315=3^25^17^1\)
\(2=2^1\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(3=3^1\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(5=5^1\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)
\(7=7^1\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(9=3^2\)		\(\checkmark\)	\(\checkmark\)		\(\checkmark\)