Partially ordered set

Elements

Partially ordered elements implements the 6 classical comparison operation:

• \(<\)

• \(\leq\)

• \(>\)

• \(\geq\)

• \(=\)

• \(\neq\)

They can be filtered using some special functions.

Integer example

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

To import the Integer class:

from galactic.examples.arithmetic.algebras import Integer
Integer(12)

$12=2^23^1$

3 is lesser than 6:

Integer(3) <= Integer(6)

True

2 and 3 are incomparable:

Integer(2) <= Integer(3)

False

Integer(3) <= Integer(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):

Integer(1) <= Integer(6)

True

Integer(6) <= Integer(0)

True

from galactic.algebras.poset.elements import top
elements = [Integer(6), Integer(24), Integer(13)]
top(elements)

<odict_iterator at 0x7fb9c7dc2570>

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

display(*top(elements))

$24=2^33^1$

$13=13^1$

An analog operation is available to get the bottom elements:

from galactic.algebras.poset.elements import bottom
display(*bottom(elements))

$6=2^13^1$

$13=13^1$

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

from galactic.algebras.poset.elements import upper_limit
display(*upper_limit(iterable=elements, limit=Integer(6)))

$6=2^13^1$

from galactic.algebras.poset.elements import lower_limit
display(
    *lower_limit(
        iterable=elements,
        limit=Integer(12),
        strict=True
    )
)

$24=2^33^1$

Partially ordered elements can be lower (and upper) bounded.

This is the case of the Integer class:

Integer.minimum()

$1$

Integer.maximum()

$0$

Color example

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

To import the Color class:

from galactic.examples.color.algebras 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. You can use either the BasicPartiallyOrderedSet class or the CompactPartiallyOrderedSet class.

Integer example

from galactic.algebras.poset.collections import \
    BasicPartiallyOrderedSet, HasseDiagram
poset = BasicPartiallyOrderedSet([
    Integer(24),
    Integer(18),
    Integer(9),
    Integer(6),
    Integer(3),
    Integer(2)
])

poset have all classical python operations on sets:

HasseDiagram(poset)

len(poset)

6

display(*poset)

$2=2^1$

$3=3^1$

$6=2^13^1$

$9=3^2$

$18=2^13^2$

$24=2^33^1$

Integer(3) in poset

True

display(
    *(
        poset &
        BasicPartiallyOrderedSet(
            [Integer(9), Integer(6), Integer(5)]
        )
    )
)

$9=3^2$

$6=2^13^1$

display(
    *(
        poset |
        BasicPartiallyOrderedSet(
            [Integer(9), Integer(6), Integer(5)]
        )
    )
)

$2=2^1$

$3=3^1$

$18=2^13^2$

$5=5^1$

$6=2^13^1$

$24=2^33^1$

$9=3^2$

poset @= BasicPartiallyOrderedSet(
    [Integer(9), Integer(6), Integer(5)]
)
HasseDiagram(poset)

poset += [Integer(10)]
HasseDiagram(poset)

poset >= BasicPartiallyOrderedSet([Integer(9), Integer(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 siblings, co-parents or neighbors (sibling or co-parent) of an element.

display(*poset.top())

$10=2^15^1$

$18=2^13^2$

$24=2^33^1$

display(*poset.bottom())

$2=2^1$

$3=3^1$

$5=5^1$

display(*poset.descendants(Integer(18)))

display(*poset.ascendants(Integer(6)))

$2=2^1$

$3=3^1$

display(*poset.successors(Integer(6)))

$24=2^33^1$

$18=2^13^2$

display(*poset.predecessors(Integer(6)))

$2=2^1$

$3=3^1$

display(*poset.siblings(Integer(6)))

$9=3^2$

$10=2^15^1$

Color example

Using colors, you can can test the poset collections:

from galactic.algebras.poset.collections import \
    CompactPartiallyOrderedSet
poset = CompactPartiallyOrderedSet([
    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(poset)

len(poset)

6

display(*poset)

poset >= CompactPartiallyOrderedSet([
    Color(green=1.0, blue=0.5),
    Color(green=1.0, blue=1.0)
])

True

display(*poset.top())

display(*poset.bottom())

Color(red=0.5)

display(*poset.descendants(Color(red=0.5)))

Color(red=1.0)

display(*poset.ascendants(Color(red=1.0)))

display(*poset.successors(Color(red=0.5)))

display(*poset.predecessors(Color(red=1.0)))