Partially ordered set

Elements

Partially ordered elements implements the 6 classical comparison operation:

• \(<\)

• \(\leq\)

• \(>\)

• \(\geq\)

• \(=\)

• \(\neq\)

They can be filtered using some special functions.

PrimeFactors example

The PrimeFactors class in the examples folder implements a partially ordered relation between positive integers using the divisor of notion.

To import the PrimeFactors class:

from galactic.algebras.examples.arithmetic import PrimeFactors

PrimeFactors(12)

$12=2^2·3$

3 is lesser than 6:

PrimeFactors(3) <= PrimeFactors(6)

True

2 and 3 are incomparable:

PrimeFactors(2) <= PrimeFactors(3)

False

PrimeFactors(3) <= PrimeFactors(2)

False

1 is lesser than all the other numbers (1 is a divisor of all numbers) and 0 is greater than all other numbers (all numbers are divisors of 0):

PrimeFactors(1) <= PrimeFactors(6)

True

PrimeFactors(6) <= PrimeFactors(0)

True

from galactic.algebras.poset import top

elements = [PrimeFactors(6), PrimeFactors(24), PrimeFactors(13)]
top(elements)

<odict_iterator at 0x70bffc770fe0>

Note that the result is a python iterator. To get the list:

display(*top(elements))

$24=2^3·3$

$13=13$

An analog operation is available to get the bottom elements:

from galactic.algebras.poset import bottom

display(*bottom(elements))

$6=2·3$

$13=13$

It’s possible to get the elements greater or lower than a given limit:

from galactic.algebras.poset import upper_limit

display(*upper_limit(iterable=elements, limit=PrimeFactors(6)))

$6=2·3$

from galactic.algebras.poset import lower_limit

display(*lower_limit(iterable=elements, limit=PrimeFactors(12), strict=True))

$24=2^3·3$

Color example

The Color class in the examples folder implements a partially ordered relation between colors.

To import the Color class:

from galactic.algebras.examples.color import Color

Color(red=1.0, green=0.5)

Color(red=1.0, green=1.0)

The orange color is less or equal than the yellow color:

Color(red=1.0, green=0.5) <= Color(red=1.0, green=1.0)

True

Color(red=0.5, green=1.0)

Collections

A poset is a set or partially ordered elements.

PrimeFactors example

from galactic.algebras.examples.arithmetic.renderer import PrimeFactorsRenderer
from galactic.algebras.poset import MutableFinitePartiallyOrderedSet
from galactic.algebras.poset.renderer import HasseDiagram

poset = MutableFinitePartiallyOrderedSet[PrimeFactors](
    elements=[
        PrimeFactors(24),
        PrimeFactors(18),
        PrimeFactors(9),
        PrimeFactors(6),
        PrimeFactors(3),
        PrimeFactors(2),
    ],
)

poset have all classical python operations on sets:

HasseDiagram[PrimeFactors](poset, domain_renderer=PrimeFactorsRenderer())

len(poset)

6

display(*poset.cover)

(PrimeFactors(3), PrimeFactors(9))

(PrimeFactors(3), PrimeFactors(6))

(PrimeFactors(2), PrimeFactors(6))

(PrimeFactors(9), PrimeFactors(18))

(PrimeFactors(6), PrimeFactors(24))

(PrimeFactors(6), PrimeFactors(18))

PrimeFactors(3) in poset.cover.domain

True

poset.extend([PrimeFactors(9), PrimeFactors(6), PrimeFactors(5)])
HasseDiagram[PrimeFactors](poset, domain_renderer=PrimeFactorsRenderer())

poset.extend([PrimeFactors(10)])
HasseDiagram[PrimeFactors](poset, domain_renderer=PrimeFactorsRenderer())

poset >= {PrimeFactors(9), PrimeFactors(6)}

True

From a poset, several additional methods can be applied:

• get the top or the bottom elements from the poset;

• get the descendants or the ascendants of an element;

• get the successors and predecessors of an element;

• get the filter and an ideal of an element.

display(*poset.top)

$10=2·5$

$18=2·3^2$

$24=2^3·3$

display(*poset.bottom)

$3=3$

$2=2$

$5=5$

display(*poset.cover.predecessors(PrimeFactors(18)))

$9=3^2$

$6=2·3$

display(*poset.order.predecessors(PrimeFactors(18)))

$9=3^2$

$6=2·3$

$3=3$

$2=2$

$18=2·3^2$

display(*poset.cover.successors(PrimeFactors(6)))

$24=2^3·3$

$18=2·3^2$

display(*poset.order.successors(PrimeFactors(6)))

$6=2·3$

$24=2^3·3$

$18=2·3^2$

HasseDiagram[PrimeFactors](poset, domain_renderer=PrimeFactorsRenderer())

HasseDiagram[PrimeFactors](
    poset.ideal(PrimeFactors(18)),
    domain_renderer=PrimeFactorsRenderer(),
)

HasseDiagram[PrimeFactors](
    poset.filter(PrimeFactors(3)),
    domain_renderer=PrimeFactorsRenderer(),
)

HasseDiagram[PrimeFactors](
    poset.filter(PrimeFactors(3)).ideal(PrimeFactors(18)),
    domain_renderer=PrimeFactorsRenderer(),
)

HasseDiagram[PrimeFactors](
    poset.ideal(PrimeFactors(18)).filter(PrimeFactors(3)),
    domain_renderer=PrimeFactorsRenderer(),
)

Color example

Using colors, you can can test the poset collections:

from galactic.algebras.examples.color.renderer import ColorRenderer
from galactic.algebras.poset import MutableFinitePartiallyOrderedSet

colors = MutableFinitePartiallyOrderedSet[Color](
    elements=[
        Color(red=1.0),
        Color(red=1.0, green=1.0),
        Color(red=0.5),
        Color(red=0.5, green=0.5),
        Color(green=1.0, blue=0.5),
        Color(green=1.0, blue=1.0),
    ],
)

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

len(colors)

6

display(*colors)

display(*colors.cover)

(Color(red=0.5, green=0.0, blue=0.0), Color(red=1.0, green=0.0, blue=0.0))

(Color(red=0.5, green=0.0, blue=0.0), Color(red=0.5, green=0.5, blue=0.0))

(Color(red=0.0, green=1.0, blue=0.5), Color(red=0.0, green=1.0, blue=1.0))

(Color(red=1.0, green=0.0, blue=0.0), Color(red=1.0, green=1.0, blue=0.0))

(Color(red=0.5, green=0.5, blue=0.0), Color(red=1.0, green=1.0, blue=0.0))

colors.cover.domain >= {Color(green=1.0, blue=0.5), Color(green=1.0, blue=1.0)}

True

display(*colors.top)

display(*colors.bottom)

Color(red=0.5)

display(*colors.cover.successors(Color(red=0.5)))

display(*colors.order.successors(Color(red=0.5)))

Color(red=1.0)

display(*colors.cover.predecessors(Color(red=1.0)))

display(*colors.order.predecessors(Color(red=1.0)))