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:

[1]:

from galactic.examples.arithmetic import Integer

Integer(12)

[1]:

$12=2^23^1$

3 is lesser than 6:

[2]:

Integer(3) <= Integer(6)

[2]:

True

2 and 3 are incomparable:

[3]:

Integer(2) <= Integer(3)

[3]:

False

[4]:

Integer(3) <= Integer(2)

[4]:

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):

[5]:

Integer(1) <= Integer(6)

[5]:

True

[6]:

Integer(6) <= Integer(0)

[6]:

True

[7]:

from galactic.algebras.poset import top

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

[7]:

<odict_iterator at 0x7f13def2c360>

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

[8]:

display(*top(elements))

$24=2^33^1$

$13=13^1$

An analog operation is available to get the bottom elements:

[9]:

from galactic.algebras.poset 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:

[10]:

from galactic.algebras.poset import upper_limit

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

$6=2^13^1$

[11]:

from galactic.algebras.poset import lower_limit

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

$24=2^33^1$

Color example

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

To import the Color class:

[12]:

from galactic.examples.color import Color

[13]:

Color(red=1.0, green=0.5)

[13]:

../../../_images/notebooks_algebras_poset_notebook_20_0.svg

[14]:

Color(red=1.0, green=1.0)

[14]:

../../../_images/notebooks_algebras_poset_notebook_21_0.svg

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

[15]:

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

[15]:

True

[16]:

Color(red=0.5, green=1.0)

[16]:

../../../_images/notebooks_algebras_poset_notebook_24_0.svg

Collections

A poset is a set or partially ordered elements.

Integer example

[17]:

from galactic.algebras.poset import PartiallyOrderedSet, HasseDiagram
from galactic.examples.arithmetic import IntegerRenderer

poset = PartiallyOrderedSet[Integer](
    domain=[Integer(24), Integer(18), Integer(9), Integer(6), Integer(3), Integer(2)]
)

poset have all classical python operations on sets:

[18]:

HasseDiagram[Integer](poset, domain_renderer=IntegerRenderer())

[18]:

../../../_images/notebooks_algebras_poset_notebook_28_0.png

[19]:

len(poset)

[19]:

[20]:

display(*poset.cover)

(Integer(2), Integer(6))

(Integer(3), Integer(9))

(Integer(3), Integer(6))

(Integer(6), Integer(24))

(Integer(6), Integer(18))

(Integer(9), Integer(18))

[21]:

Integer(3) in poset.domain

[21]:

True

[22]:

display(*(poset.domain & {Integer(9), Integer(6), Integer(5)}))

$9=3^2$

$6=2^13^1$

[23]:

display(*(poset.domain | {Integer(9), Integer(6), Integer(5)}))

$24=2^33^1$

$18=2^13^2$

$2=2^1$

$3=3^1$

$6=2^13^1$

$9=3^2$

$5=5^1$

[24]:

poset.extend([Integer(9), Integer(6), Integer(5)])
HasseDiagram[Integer](poset, domain_renderer=IntegerRenderer())

[24]:

../../../_images/notebooks_algebras_poset_notebook_34_0.png

[25]:

poset.extend([Integer(10)])
HasseDiagram[Integer](poset, domain_renderer=IntegerRenderer())

[25]:

../../../_images/notebooks_algebras_poset_notebook_35_0.png

[26]:

poset.domain >= {Integer(9), Integer(6)}

[26]:

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.

[27]:

display(*poset.top)

$24=2^33^1$

$18=2^13^2$

$10=2^15^1$

[28]:

display(*poset.bottom)

$3=3^1$

$2=2^1$

$5=5^1$

[29]:

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

$18=2^13^2$

$2=2^1$

$3=3^1$

$6=2^13^1$

$9=3^2$

[30]:

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

$9=3^2$

$6=2^13^1$

[31]:

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

$24=2^33^1$

$18=2^13^2$

$6=2^13^1$

[32]:

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

$24=2^33^1$

$18=2^13^2$

[33]:

HasseDiagram[Integer](poset, domain_renderer=IntegerRenderer())

[33]:

../../../_images/notebooks_algebras_poset_notebook_44_0.png

[34]:

HasseDiagram[Integer](poset.ideal(Integer(18)), domain_renderer=IntegerRenderer())

[34]:

../../../_images/notebooks_algebras_poset_notebook_45_0.png

[35]:

HasseDiagram[Integer](poset.filter(Integer(3)), domain_renderer=IntegerRenderer())

[35]:

../../../_images/notebooks_algebras_poset_notebook_46_0.png

[36]:

HasseDiagram[Integer](
    poset.filter(Integer(3)).ideal(Integer(18)), domain_renderer=IntegerRenderer()
)

[36]:

../../../_images/notebooks_algebras_poset_notebook_47_0.png

[37]:

HasseDiagram[Integer](
    poset.ideal(Integer(18)).filter(Integer(3)), domain_renderer=IntegerRenderer()
)

[37]:

../../../_images/notebooks_algebras_poset_notebook_48_0.png

Color example

Using colors, you can can test the poset collections:

[38]:

from galactic.algebras.poset import PartiallyOrderedSet
from galactic.examples.color import ColorRenderer

colors = PartiallyOrderedSet[Color](
    domain=[
        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),
    ]
)

[39]:

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

[39]:

../../../_images/notebooks_algebras_poset_notebook_51_0.png

[40]:

len(colors)

[40]:

[41]:

display(*colors)

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

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

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

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

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

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

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

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

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

[42]:

display(*colors.cover)

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

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

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

[43]:

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

[43]:

True

[44]:

display(*colors.top)

../../../_images/notebooks_algebras_poset_notebook_56_0.svg

../../../_images/notebooks_algebras_poset_notebook_56_1.svg

[45]:

display(*colors.bottom)

../../../_images/notebooks_algebras_poset_notebook_57_0.svg

../../../_images/notebooks_algebras_poset_notebook_57_1.svg

[46]:

Color(red=0.5)

[46]:

../../../_images/notebooks_algebras_poset_notebook_58_0.svg

[47]:

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

../../../_images/notebooks_algebras_poset_notebook_59_0.svg

../../../_images/notebooks_algebras_poset_notebook_59_1.svg

../../../_images/notebooks_algebras_poset_notebook_59_2.svg

../../../_images/notebooks_algebras_poset_notebook_59_3.svg

[48]:

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

../../../_images/notebooks_algebras_poset_notebook_60_0.svg

../../../_images/notebooks_algebras_poset_notebook_60_1.svg

[49]:

Color(red=1.0)

[49]:

../../../_images/notebooks_algebras_poset_notebook_61_0.svg

[50]:

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

../../../_images/notebooks_algebras_poset_notebook_62_0.svg

../../../_images/notebooks_algebras_poset_notebook_62_1.svg

[51]:

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

../../../_images/notebooks_algebras_poset_notebook_63_0.svg