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:

[1]:

from galactic.algebras.examples.arithmetic import PrimeFactors

PrimeFactors(12)

[1]:

$12=2^2·3$

3 is lesser than 6:

[2]:

PrimeFactors(3) <= PrimeFactors(6)

[2]:

True

2 and 3 are incomparable:

[3]:

PrimeFactors(2) <= PrimeFactors(3)

[3]:

False

[4]:

PrimeFactors(3) <= PrimeFactors(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]:

PrimeFactors(1) <= PrimeFactors(6)

[5]:

True

[6]:

PrimeFactors(6) <= PrimeFactors(0)

[6]:

True

[7]:

from galactic.algebras.poset import top

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

[7]:

<odict_iterator at 0x7f3d2c603d30>

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

[8]:

display(*top(elements))

$24=2^3·3$

$13=13$

An analog operation is available to get the bottom elements:

[9]:

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:

[10]:

from galactic.algebras.poset import upper_limit

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

$6=2·3$

[11]:

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:

[12]:

from galactic.algebras.examples.color import Color

[13]:

Color(red=1.0, green=0.5)

[13]:

../../_images/notebooks_poset_notebook_20_0.svg

[14]:

Color(red=1.0, green=1.0)

[14]:

../../_images/notebooks_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_poset_notebook_24_0.svg

Collections

A poset is a set or partially ordered elements.

PrimeFactors example

[17]:

from galactic.algebras.poset import FinitePartiallyOrderedSet, HasseDiagram
from galactic.algebras.examples.arithmetic import PrimeFactorsRenderer

poset = FinitePartiallyOrderedSet[PrimeFactors](
    domain=[
        PrimeFactors(24),
        PrimeFactors(18),
        PrimeFactors(9),
        PrimeFactors(6),
        PrimeFactors(3),
        PrimeFactors(2),
    ]
)

poset have all classical python operations on sets:

[18]:

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

[18]:

../../_images/notebooks_poset_notebook_28_0.png

[19]:

len(poset)

[19]:

[20]:

display(*poset.cover)

(PrimeFactors(2), PrimeFactors(6))

(PrimeFactors(3), PrimeFactors(9))

(PrimeFactors(3), PrimeFactors(6))

(PrimeFactors(6), PrimeFactors(24))

(PrimeFactors(6), PrimeFactors(18))

(PrimeFactors(9), PrimeFactors(18))

[21]:

PrimeFactors(3) in poset.domain

[21]:

True

[22]:

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

$9=3^2$

$6=2·3$

[23]:

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

$24=2^3·3$

$18=2·3^2$

$2=2$

$3=3$

$6=2·3$

$9=3^2$

$5=5$

[24]:

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

[24]:

../../_images/notebooks_poset_notebook_34_0.png

[25]:

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

[25]:

../../_images/notebooks_poset_notebook_35_0.png

[26]:

poset.domain >= {PrimeFactors(9), PrimeFactors(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)

$24=2^3·3$

$18=2·3^2$

$10=2·5$

[28]:

display(*poset.bottom)

$3=3$

$2=2$

$5=5$

[29]:

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

$18=2·3^2$

$2=2$

$3=3$

$6=2·3$

$9=3^2$

[30]:

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

$9=3^2$

$6=2·3$

[31]:

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

$24=2^3·3$

$18=2·3^2$

$6=2·3$

[32]:

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

$24=2^3·3$

$18=2·3^2$

[33]:

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

[33]:

../../_images/notebooks_poset_notebook_44_0.png

[34]:

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

[34]:

../../_images/notebooks_poset_notebook_45_0.png

[35]:

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

[35]:

../../_images/notebooks_poset_notebook_46_0.png

[36]:

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

[36]:

../../_images/notebooks_poset_notebook_47_0.png

[37]:

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

[37]:

../../_images/notebooks_poset_notebook_48_0.png

Color example

Using colors, you can can test the poset collections:

[38]:

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

colors = FinitePartiallyOrderedSet[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_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_poset_notebook_56_0.svg

../../_images/notebooks_poset_notebook_56_1.svg

[45]:

display(*colors.bottom)

../../_images/notebooks_poset_notebook_57_0.svg

../../_images/notebooks_poset_notebook_57_1.svg

[46]:

Color(red=0.5)

[46]:

../../_images/notebooks_poset_notebook_58_0.svg

[47]:

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

../../_images/notebooks_poset_notebook_59_0.svg

../../_images/notebooks_poset_notebook_59_1.svg

../../_images/notebooks_poset_notebook_59_2.svg

../../_images/notebooks_poset_notebook_59_3.svg

[48]:

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

../../_images/notebooks_poset_notebook_60_0.svg

../../_images/notebooks_poset_notebook_60_1.svg

[49]:

Color(red=1.0)

[49]:

../../_images/notebooks_poset_notebook_61_0.svg

[50]:

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

../../_images/notebooks_poset_notebook_62_0.svg

../../_images/notebooks_poset_notebook_62_1.svg

[51]:

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

../../_images/notebooks_poset_notebook_63_0.svg