Binary Relation

MutableFiniteBinaryRelation can be used to store a mutable binary relation. You can use FrozenFiniteBinaryRelation for frozen binary relation.

[1]:

from galactic.algebras.relational import MutableFiniteBinaryRelation
from galactic.algebras.relational.renderer import SagittalDiagram

It can be initialized using an iterable of couples.

[2]:

relation = MutableFiniteBinaryRelation[int, int](
    (1, 2, 3),
    (1, 2, 3),
    elements=[(1, 2), (1, 3), (2, 3)],
)
SagittalDiagram(relation)

[2]:
../../../_images/notebooks_algebras_relational_notebook_3_0.png 
[3]:

list(relation)

[3]:

[(1, 2), (1, 3), (2, 3)]

Binary relation are considered as sets.

[4]:

len(relation)

[4]:

3

They have a domain and a co-domain also accessible by a tuple of universes.

[5]:

list(relation.domain)

[5]:

[1, 2, 3]

[6]:

list(relation.universes[0])

[6]:

[1, 2, 3]

[7]:

list(relation.co_domain)

[7]:

[1, 2, 3]

[8]:

list(relation.universes[1])

[8]:

[1, 2, 3]

[9]:

relation.add((2, 1))
SagittalDiagram(relation)

[9]:
../../../_images/notebooks_algebras_relational_notebook_12_0.png 
[10]:

list(relation)

[10]:

[(1, 2), (1, 3), (2, 3), (2, 1)]

[11]:

list(relation.domain)

[11]:

[1, 2, 3]

[12]:

list(relation.co_domain)

[12]:

[1, 2, 3]

Successors and predecessors of an element are considered as sets. They are equivalent to a combination of a selection and a projection.

[13]:

relation.successors(1)

[13]:

<MutableSuccessors object at 0x741d41510f80>

[14]:

list(relation.successors(1))

[14]:

[2, 3]

[15]:

relation.successors(1).add(1)
SagittalDiagram(relation)

[15]:
../../../_images/notebooks_algebras_relational_notebook_19_0.png 
[16]:

list(relation)

[16]:

[(1, 2), (1, 3), (1, 1), (2, 3), (2, 1)]

[17]:

relation.successors(1).discard(3)
SagittalDiagram(relation)

[17]:
../../../_images/notebooks_algebras_relational_notebook_21_0.png 
[18]:

list(relation)

[18]:

[(1, 2), (1, 1), (2, 3), (2, 1)]

[19]:

relation.domain.add(4)
list(relation.domain)

[19]:

[1, 2, 3, 4]

[20]:

list(relation.co_domain)

[20]:

[1, 2, 3]

[21]:

relation.successors(2).clear()
SagittalDiagram(relation)

[21]:
../../../_images/notebooks_algebras_relational_notebook_25_0.png 
[22]:

list(relation)

[22]:

[(1, 2), (1, 1)]

[23]:

list(relation.predecessors(2))

[23]:

[1]

[24]:

predecessors = relation.predecessors(2)
predecessors.extend({2, 3, 4})
SagittalDiagram(relation)

[24]:
../../../_images/notebooks_algebras_relational_notebook_28_0.png 
[25]:

list(relation)

[25]:

[(1, 2), (1, 1), (2, 2), (3, 2), (4, 2)]

[26]:

relation.domain.discard(1)
SagittalDiagram(relation)

[26]:
../../../_images/notebooks_algebras_relational_notebook_30_0.png 
[27]:

list(relation)

[27]:

[(2, 2), (3, 2), (4, 2)]

Endo Relation

MutableFiniteEndoRelation can be used to store a mutable endo relation. You can use FrozenFiniteEndoRelation for frozen endo relation.

[28]:

from galactic.algebras.relational import MutableFiniteEndoRelation
from galactic.algebras.relational.renderer import EndoDiagram, SagittalDiagram

It can be initialized using an iterable of couples.

[29]:

relation = MutableFiniteEndoRelation[int, int](
    (1, 2, 3),
    elements=[(1, 2), (1, 3), (2, 3)],
)
EndoDiagram(relation)

[29]:
../../../_images/notebooks_algebras_relational_notebook_35_0.png 
[30]:

SagittalDiagram(relation)

[30]:
../../../_images/notebooks_algebras_relational_notebook_36_0.png

All operations that are possible on binary relations are possible on endo relation.

[31]:

list(relation)

[31]:

[(1, 2), (1, 3), (2, 3)]

[32]:

list(relation.domain)

[32]:

[1, 2, 3]

[33]:

list(relation.co_domain)

[33]:

[1, 2, 3]

[34]:

relation.successors(1)

[34]:

<MutableSuccessors object at 0x741d41511d30>

[35]:

list(relation.successors(1))

[35]:

[2, 3]

[36]:

relation.successors(1).add(1)
EndoDiagram(relation)

[36]:
../../../_images/notebooks_algebras_relational_notebook_43_0.png 
[37]:

list(relation)

[37]:

[(1, 2), (1, 3), (1, 1), (2, 3)]

[38]:

relation.successors(1).discard(3)
EndoDiagram(relation)

[38]:
../../../_images/notebooks_algebras_relational_notebook_45_0.png 
[39]:

relation.domain.add(4)
list(relation.domain)

[39]:

[1, 2, 3, 4]

[40]:

list(relation.co_domain)

[40]:

[1, 2, 3, 4]

[41]:

EndoDiagram(relation)

[41]:
../../../_images/notebooks_algebras_relational_notebook_48_0.png 
[42]:

relation.successors(1).add(4)
EndoDiagram(relation)

[42]:
../../../_images/notebooks_algebras_relational_notebook_49_0.png 
[43]:

relation.successors(4).add(3)
EndoDiagram(relation)

[43]:
../../../_images/notebooks_algebras_relational_notebook_50_0.png 
[44]:

list(relation)

[44]:

[(1, 2), (1, 1), (1, 4), (2, 3), (4, 3)]

[45]:

list(relation.successors(1))

[45]:

[2, 1, 4]

[46]:

list(relation.successors(2))

[46]:

[3]

[47]:

list(relation.successors(3))

[47]:

[]

[48]:

list(relation.successors(4))

[48]:

[3]

[49]:

list(relation.predecessors(1))

[49]:

[1]

[50]:

list(relation.predecessors(2))

[50]:

[1]

[51]:

list(relation.predecessors(3))

[51]:

[2, 4]

[52]:

list(relation.predecessors(4))

[52]:

[1]

[53]:

relation.co_domain.remove(3)
EndoDiagram(relation)

[53]:
../../../_images/notebooks_algebras_relational_notebook_60_0.png 
[54]:

list(relation.domain)

[54]:

[1, 2, 4]

[55]:

list(relation.co_domain)

[55]:

[1, 2, 4]

[56]:

relation.pop()

[56]:

(1, 2)

[57]:

EndoDiagram(relation)

[57]:
../../../_images/notebooks_algebras_relational_notebook_64_0.png 
[58]:

relation.pop()

[58]:

(1, 1)

[59]:

EndoDiagram(relation)

[59]:
../../../_images/notebooks_algebras_relational_notebook_66_0.png 
[60]:

relation.pop()

[60]:

(1, 4)

[61]:

EndoDiagram(relation)

[61]:
../../../_images/notebooks_algebras_relational_notebook_68_0.png