Closures

Closure operators

In mathematics, a closure operator \(\phi\) on a set \(S\) is a function from the power set of \(S\) to itself which is extensive, increasing and idempotent: \(\phi:2^S\rightarrow 2^S\)

such that:

  • \(X \subseteq \phi(X) \subseteq S\)

  • \(X \subseteq Y \Rightarrow \phi(X) \subseteq \phi(Y)\)

  • \(\phi(\phi(X))=\phi(X)\)

Applying a closure operator

[1]:

from galactic.algebras.examples.closure import (
    ExtensibleNumericalFamily,
    NumericalClosure,
)

[2]:

closure = NumericalClosure(min_element=0, max_element=9)

[3]:

closure(elements={0, 1, 4})

[3]:

0..4

[4]:

list(closure(elements={0, 1, 4}))

[4]:

[0, 1, 2, 3, 4]

[5]:

a = closure(elements={3, 5})
b = closure(elements={2, 4})

[6]:

a

[6]:

3..5

[7]:

b

[7]:

2..4

[8]:

closure(a, b)

[8]:

2..5

[9]:

closure(a, b, elements={7})

[9]:

2..7

Get the universe \(S\)

[10]:

list(closure())

[10]:

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

[11]:

0 in closure()

[11]:

True

[12]:

20 in closure()

[12]:

False

[13]:

len(closure())

[13]:

10

Closure systems (Moore family)

In mathematics, a Moore family (or a closure system) is a family of subsets of a set \(S\) containing \(S\) and closed by set intersection. It can be defined by a closure operator. They can be seen as lattices of closed sets and in this sense, our Moore family definition class creates sub-lattices of the Moore family that would have been generated with the initial closure operator.

Defining a Moore family

[14]:

family = ExtensibleNumericalFamily(closure)

[15]:

family.extend(
    [
        closure(elements={1, 2, 3}),
        closure(elements={2, 3, 4}),
    ],
)

[16]:

list(family)

[16]:

[0..9, 1..4, 1..3, 2..4, 2..3]

Drawing the Hasse diagram of a Moore family

[17]:

from galactic.algebras.poset.renderer import HasseDiagram

[18]:

HasseDiagram(family)

[18]:
../../../_images/notebooks_algebras_closure_notebook_21_0.png

Note the presence of the \(1..4\) closed set which is the result of the join between \(1..3\) and \(2..4\): \(1..3 \vee 2..4\) (this Moore family is a sub-lattice of the global Moore family generated from the closure operator).

Getting the underlying closure operator of a Moore family

[19]:

family_closure = family.closure()

[20]:

family_closure(elements={1})

[20]:

1..3

[21]:

closure(elements={1})

[21]:

1..1

[22]:

family.extend([closure(elements={1})])

[23]:

HasseDiagram(family)

[23]:
../../../_images/notebooks_algebras_closure_notebook_28_0.png 
[24]:

family_closure(elements={1})

[24]:

1..3

[25]:

new_family_closure = family.closure()

[26]:

new_family_closure(elements={1})

[26]:

1..1

Measuring a Moore family

[27]:

family.global_stability

[27]:

0.9766845703125

[28]:

family.global_logarithmic_stability

[28]:

0.586037538336511

[29]:

family.global_information

[29]:

1.5370318565210717

[30]:

family.global_entropy

[30]:

0.15370318565210717

[31]:

for stability in family.logarithmic_stabilities():
    print(stability.closed, round(stability.value, 3))

∅ 1.0
2..3 0.5
1..1 0.0
2..4 0.226
1..3 0.138
1..4 0.075
0..9 0.591