Posets
The galactic.algebras.poset
module defines types for representing posets.
their implementations:
and their elements which are partially comparable:
with their functions:
It also defines the HasseDiagram
class for drawing Hasse diagram of
partially ordered sets into jupyter notebooks.
- class AbstractCoveringRelation(*args, **kwds)
The
AbstractCoveringRelation
represents the covering relation of a poset.- property co_domain: AbstractSet[galactic.algebras.relational._binary._F]
Get the co-domain of this binary relation.
This is a proxy to
self.universes[1]
.- Returns:
The co-domain of this binary relation.
- Return type:
- property domain: AbstractSet[galactic.algebras.relational._binary._E]
Get the domain of this binary relation.
This is a proxy to
self.universes[0]
.- Returns:
The domain of this binary relation.
- Return type:
- isdisjoint(other)
Return True if two sets have a null intersection.
- abstract neighbours() Iterable[Tuple[galactic.algebras.poset._abstract._P, AbstractSet[galactic.algebras.poset._abstract._P], AbstractSet[galactic.algebras.poset._abstract._P]]]
Get an iterable over the neighbours of elements.
Each iteration yields a triple
element, successors(element), predecessors(element)
.- Returns:
- Return type:
- abstract predecessors(element: galactic.algebras.relational._binary._F) AbstractSet[galactic.algebras.relational._binary._E]
Get the predecessors of an
element
.- Parameters:
element (
_F
) – The element whose predecessors are requested- Returns:
The predecessors.
- Return type:
- Raises:
ValueError – If the element does not belong to the relation.
- abstract property sinks: AbstractSet[galactic.algebras.poset._abstract._P]
Get the elements with no successors.
- Returns:
The elements with no successors.
- Return type:
- abstract property sources: AbstractSet[galactic.algebras.poset._abstract._P]
Get the elements with no predecessors.
- Returns:
The elements with no predecessors.
- Return type:
- abstract successors(element: galactic.algebras.relational._binary._E) AbstractSet[galactic.algebras.relational._binary._F]
Get the successors of an
element
.- Parameters:
element (
_E
) – The element whose successors are requested- Returns:
The successors.
- Return type:
- Raises:
ValueError – If the element does not belong to the relation.
- abstract property universes: Tuple[AbstractSet[galactic.algebras.relational._binary._E], AbstractSet[galactic.algebras.relational._binary._F]]
Get the universes of this relation.
- Returns:
The universes of this relation.
- Return type:
- class AbstractReversibleCoveringRelation(*args, **kwds)
The
AbstractReversibleCoveringRelation
class.It represents a reversible version of the covering relation of a poset.
- property co_domain: AbstractSet[galactic.algebras.relational._binary._F]
Get the co-domain of this binary relation.
This is a proxy to
self.universes[1]
.- Returns:
The co-domain of this binary relation.
- Return type:
- property domain: AbstractSet[galactic.algebras.relational._binary._E]
Get the domain of this binary relation.
This is a proxy to
self.universes[0]
.- Returns:
The domain of this binary relation.
- Return type:
- isdisjoint(other)
Return True if two sets have a null intersection.
- abstract neighbours() Reversible[Tuple[galactic.algebras.poset._abstract._P, AbstractSet[galactic.algebras.poset._abstract._P], AbstractSet[galactic.algebras.poset._abstract._P]]]
Get a reversible iterable over the neighbours of elements.
Each iteration yields a triple
element, successors(element), predecessors(element)
.- Returns:
- Return type:
- abstract predecessors(element: galactic.algebras.relational._binary._F) AbstractSet[galactic.algebras.relational._binary._E]
Get the predecessors of an
element
.- Parameters:
element (
_F
) – The element whose predecessors are requested- Returns:
The predecessors.
- Return type:
- Raises:
ValueError – If the element does not belong to the relation.
- abstract property sinks: AbstractSet[galactic.algebras.poset._abstract._P]
Get the elements with no successors.
- Returns:
The elements with no successors.
- Return type:
- abstract property sources: AbstractSet[galactic.algebras.poset._abstract._P]
Get the elements with no predecessors.
- Returns:
The elements with no predecessors.
- Return type:
- abstract successors(element: galactic.algebras.relational._binary._E) AbstractSet[galactic.algebras.relational._binary._F]
Get the successors of an
element
.- Parameters:
element (
_E
) – The element whose successors are requested- Returns:
The successors.
- Return type:
- Raises:
ValueError – If the element does not belong to the relation.
- abstract property universes: Tuple[AbstractSet[galactic.algebras.relational._binary._E], AbstractSet[galactic.algebras.relational._binary._F]]
Get the universes of this relation.
- Returns:
The universes of this relation.
- Return type:
- class AbstractPartiallyOrderedSet(*args, **kwds)
The :class`AbstractPartiallyOrderedSet` abstract class.
A partially ordered set is an endo-relation over a set of
PartiallyComparable
elements.- __iter__() Iterator[Tuple[galactic.algebras.poset._abstract._P, galactic.algebras.poset._abstract._P]]
Get an iterator over the elements.
- Returns:
An iterator over the elements
- Return type:
- abstract property bottom: AbstractSet[galactic.algebras.poset._abstract._P]
Get the bottom elements.
- Returns:
The bottom elements.
- Return type:
- property co_domain: AbstractSet[galactic.algebras.relational._binary._F]
Get the co-domain of this binary relation.
This is a proxy to
self.universes[1]
.- Returns:
The co-domain of this binary relation.
- Return type:
- abstract property cover: galactic.algebras.poset.AbstractCoveringRelation[galactic.algebras.poset._abstract._P]
Get the covering relation.
- Returns:
The covering relation.
- Return type:
- property domain: AbstractSet[galactic.algebras.relational._binary._E]
Get the domain of this binary relation.
This is a proxy to
self.universes[0]
.- Returns:
The domain of this binary relation.
- Return type:
- abstract filter(element: galactic.algebras.poset._abstract._P) galactic.algebras.poset.AbstractLowerBoundedSet[galactic.algebras.poset._abstract._P]
Get a filter of a poset.
- Parameters:
element (
_P
) – The lower limit- Returns:
The lower bounded poset.
- Return type:
- Raises:
ValueError – If the element does not belong to the poset.
- abstract ideal(element: galactic.algebras.poset._abstract._P) galactic.algebras.poset.AbstractUpperBoundedSet[galactic.algebras.poset._abstract._P]
Get an ideal of the poset.
- Parameters:
element (
_P
) – The upper limit- Returns:
The upper bounded set.
- Return type:
- isdisjoint(other)
Return True if two sets have a null intersection.
- abstract predecessors(element: galactic.algebras.relational._binary._F) AbstractSet[galactic.algebras.relational._binary._E]
Get the predecessors of an
element
.- Parameters:
element (
_F
) – The element whose predecessors are requested- Returns:
The predecessors.
- Return type:
- Raises:
ValueError – If the element does not belong to the relation.
- abstract property sinks: AbstractSet[galactic.algebras.relational._binary._E]
Get the sinks of this DAG.
- Returns:
The sinks of this DAG.
- Return type:
- abstract property sources: AbstractSet[galactic.algebras.relational._binary._E]
Get the sources of this DAG.
- Returns:
The sources of this DAG.
- Return type:
- abstract successors(element: galactic.algebras.relational._binary._E) AbstractSet[galactic.algebras.relational._binary._F]
Get the successors of an
element
.- Parameters:
element (
_E
) – The element whose successors are requested- Returns:
The successors.
- Return type:
- Raises:
ValueError – If the element does not belong to the relation.
- abstract property top: AbstractSet[galactic.algebras.poset._abstract._P]
Get the top elements.
- Returns:
The top elements.
- Return type:
- abstract property universes: Tuple[AbstractSet[galactic.algebras.relational._binary._E], AbstractSet[galactic.algebras.relational._binary._F]]
Get the universes of this relation.
- Returns:
The universes of this relation.
- Return type:
- class AbstractUpperBoundedSet(*args, **kwds)
The
AbstractUpperBoundedSet
represents upper bounded posets.- __iter__() Iterator[Tuple[galactic.algebras.poset._abstract._P, galactic.algebras.poset._abstract._P]]
Get an iterator over the elements.
- Returns:
An iterator over the elements
- Return type:
- abstract property bottom: AbstractSet[galactic.algebras.poset._abstract._P]
Get the bottom elements.
- Returns:
The bottom elements.
- Return type:
- property co_domain: AbstractSet[galactic.algebras.relational._binary._F]
Get the co-domain of this binary relation.
This is a proxy to
self.universes[1]
.- Returns:
The co-domain of this binary relation.
- Return type:
- abstract property cover: galactic.algebras.poset.AbstractCoveringRelation[galactic.algebras.poset._abstract._P]
Get the covering relation.
- Returns:
The covering relation.
- Return type:
- property domain: AbstractSet[galactic.algebras.relational._binary._E]
Get the domain of this binary relation.
This is a proxy to
self.universes[0]
.- Returns:
The domain of this binary relation.
- Return type:
- abstract filter(element: galactic.algebras.poset._abstract._P) galactic.algebras.poset.AbstractLowerBoundedSet[galactic.algebras.poset._abstract._P]
Get a filter of a poset.
- Parameters:
element (
_P
) – The lower limit- Returns:
The lower bounded poset.
- Return type:
- Raises:
ValueError – If the element does not belong to the poset.
- abstract ideal(element: galactic.algebras.poset._abstract._P) galactic.algebras.poset.AbstractUpperBoundedSet[galactic.algebras.poset._abstract._P]
Get an ideal of the poset.
- Parameters:
element (
_P
) – The upper limit- Returns:
The upper bounded set.
- Return type:
- isdisjoint(other)
Return True if two sets have a null intersection.
- abstract property maximum: galactic.algebras.poset._abstract._P
Get the maximum of the poset.
- Returns:
The maximum element.
- Return type:
- abstract predecessors(element: galactic.algebras.relational._binary._F) AbstractSet[galactic.algebras.relational._binary._E]
Get the predecessors of an
element
.- Parameters:
element (
_F
) – The element whose predecessors are requested- Returns:
The predecessors.
- Return type:
- Raises:
ValueError – If the element does not belong to the relation.
- abstract property sinks: AbstractSet[galactic.algebras.relational._binary._E]
Get the sinks of this DAG.
- Returns:
The sinks of this DAG.
- Return type:
- abstract property sources: AbstractSet[galactic.algebras.relational._binary._E]
Get the sources of this DAG.
- Returns:
The sources of this DAG.
- Return type:
- abstract successors(element: galactic.algebras.relational._binary._E) AbstractSet[galactic.algebras.relational._binary._F]
Get the successors of an
element
.- Parameters:
element (
_E
) – The element whose successors are requested- Returns:
The successors.
- Return type:
- Raises:
ValueError – If the element does not belong to the relation.
- abstract property top: AbstractSet[galactic.algebras.poset._abstract._P]
Get the top elements.
- Returns:
The top elements.
- Return type:
- abstract property universes: Tuple[AbstractSet[galactic.algebras.relational._binary._E], AbstractSet[galactic.algebras.relational._binary._F]]
Get the universes of this relation.
- Returns:
The universes of this relation.
- Return type:
- class AbstractLowerBoundedSet(*args, **kwds)
The
AbstractLowerBoundedSet
represents lower bounded posets.- __iter__() Iterator[Tuple[galactic.algebras.poset._abstract._P, galactic.algebras.poset._abstract._P]]
Get an iterator over the elements.
- Returns:
An iterator over the elements
- Return type:
- abstract property bottom: AbstractSet[galactic.algebras.poset._abstract._P]
Get the bottom elements.
- Returns:
The bottom elements.
- Return type:
- property co_domain: AbstractSet[galactic.algebras.relational._binary._F]
Get the co-domain of this binary relation.
This is a proxy to
self.universes[1]
.- Returns:
The co-domain of this binary relation.
- Return type:
- abstract property cover: galactic.algebras.poset.AbstractCoveringRelation[galactic.algebras.poset._abstract._P]
Get the covering relation.
- Returns:
The covering relation.
- Return type:
- property domain: AbstractSet[galactic.algebras.relational._binary._E]
Get the domain of this binary relation.
This is a proxy to
self.universes[0]
.- Returns:
The domain of this binary relation.
- Return type:
- abstract filter(element: galactic.algebras.poset._abstract._P) galactic.algebras.poset.AbstractLowerBoundedSet[galactic.algebras.poset._abstract._P]
Get a filter of a poset.
- Parameters:
element (
_P
) – The lower limit- Returns:
The lower bounded poset.
- Return type:
- Raises:
ValueError – If the element does not belong to the poset.
- abstract ideal(element: galactic.algebras.poset._abstract._P) galactic.algebras.poset.AbstractUpperBoundedSet[galactic.algebras.poset._abstract._P]
Get an ideal of the poset.
- Parameters:
element (
_P
) – The upper limit- Returns:
The upper bounded set.
- Return type:
- isdisjoint(other)
Return True if two sets have a null intersection.
- abstract property minimum: galactic.algebras.poset._abstract._P
Get the minimum of the poset.
- Returns:
The minimum element.
- Return type:
- abstract predecessors(element: galactic.algebras.relational._binary._F) AbstractSet[galactic.algebras.relational._binary._E]
Get the predecessors of an
element
.- Parameters:
element (
_F
) – The element whose predecessors are requested- Returns:
The predecessors.
- Return type:
- Raises:
ValueError – If the element does not belong to the relation.
- abstract property sinks: AbstractSet[galactic.algebras.relational._binary._E]
Get the sinks of this DAG.
- Returns:
The sinks of this DAG.
- Return type:
- abstract property sources: AbstractSet[galactic.algebras.relational._binary._E]
Get the sources of this DAG.
- Returns:
The sources of this DAG.
- Return type:
- abstract successors(element: galactic.algebras.relational._binary._E) AbstractSet[galactic.algebras.relational._binary._F]
Get the successors of an
element
.- Parameters:
element (
_E
) – The element whose successors are requested- Returns:
The successors.
- Return type:
- Raises:
ValueError – If the element does not belong to the relation.
- abstract property top: AbstractSet[galactic.algebras.poset._abstract._P]
Get the top elements.
- Returns:
The top elements.
- Return type:
- abstract property universes: Tuple[AbstractSet[galactic.algebras.relational._binary._E], AbstractSet[galactic.algebras.relational._binary._F]]
Get the universes of this relation.
- Returns:
The universes of this relation.
- Return type:
- class AbstractBoundedSet(*args, **kwds)
The
AbstractBoundedSet
represents bounded posets.- __iter__() Iterator[Tuple[galactic.algebras.poset._abstract._P, galactic.algebras.poset._abstract._P]]
Get an iterator over the elements.
- Returns:
An iterator over the elements
- Return type:
- abstract property bottom: AbstractSet[galactic.algebras.poset._abstract._P]
Get the bottom elements.
- Returns:
The bottom elements.
- Return type:
- property co_domain: AbstractSet[galactic.algebras.relational._binary._F]
Get the co-domain of this binary relation.
This is a proxy to
self.universes[1]
.- Returns:
The co-domain of this binary relation.
- Return type:
- abstract property cover: galactic.algebras.poset.AbstractCoveringRelation[galactic.algebras.poset._abstract._P]
Get the covering relation.
- Returns:
The covering relation.
- Return type:
- property domain: AbstractSet[galactic.algebras.relational._binary._E]
Get the domain of this binary relation.
This is a proxy to
self.universes[0]
.- Returns:
The domain of this binary relation.
- Return type:
- abstract filter(element: galactic.algebras.poset._abstract._P) galactic.algebras.poset.AbstractLowerBoundedSet[galactic.algebras.poset._abstract._P]
Get a filter of a poset.
- Parameters:
element (
_P
) – The lower limit- Returns:
The lower bounded poset.
- Return type:
- Raises:
ValueError – If the element does not belong to the poset.
- abstract ideal(element: galactic.algebras.poset._abstract._P) galactic.algebras.poset.AbstractUpperBoundedSet[galactic.algebras.poset._abstract._P]
Get an ideal of the poset.
- Parameters:
element (
_P
) – The upper limit- Returns:
The upper bounded set.
- Return type:
- isdisjoint(other)
Return True if two sets have a null intersection.
- abstract property maximum: galactic.algebras.poset._abstract._P
Get the maximum of the poset.
- Returns:
The maximum element.
- Return type:
- abstract property minimum: galactic.algebras.poset._abstract._P
Get the minimum of the poset.
- Returns:
The minimum element.
- Return type:
- abstract predecessors(element: galactic.algebras.relational._binary._F) AbstractSet[galactic.algebras.relational._binary._E]
Get the predecessors of an
element
.- Parameters:
element (
_F
) – The element whose predecessors are requested- Returns:
The predecessors.
- Return type:
- Raises:
ValueError – If the element does not belong to the relation.
- abstract property sinks: AbstractSet[galactic.algebras.relational._binary._E]
Get the sinks of this DAG.
- Returns:
The sinks of this DAG.
- Return type:
- abstract property sources: AbstractSet[galactic.algebras.relational._binary._E]
Get the sources of this DAG.
- Returns:
The sources of this DAG.
- Return type:
- abstract successors(element: galactic.algebras.relational._binary._E) AbstractSet[galactic.algebras.relational._binary._F]
Get the successors of an
element
.- Parameters:
element (
_E
) – The element whose successors are requested- Returns:
The successors.
- Return type:
- Raises:
ValueError – If the element does not belong to the relation.
- abstract property top: AbstractSet[galactic.algebras.poset._abstract._P]
Get the top elements.
- Returns:
The top elements.
- Return type:
- abstract property universes: Tuple[AbstractSet[galactic.algebras.relational._binary._E], AbstractSet[galactic.algebras.relational._binary._F]]
Get the universes of this relation.
- Returns:
The universes of this relation.
- Return type:
- class PartiallyComparable(*args, **kwds)
Partially comparable elements.
A partially ordered type sets for each pair of elements \(a\), \(b\) either:
\(a \leq b\)
\(b \leq a\)
a and b are incomparable (\(a \| b\))
The relation \(\leq\) must be:
reflexive (\(a \leq a\))
transitive (\(a \leq b\) and \(b \leq c\) implies \(a \leq c\))
anti-symmetric (\(a \leq b\) and \(b \leq a\) implies \(a = b\))
A class implementing the
PartiallyComparable
abstract class must be declared by inheriting fromPartiallyComparable
and must implement the methods:Notes
The builtin classes
are considered as subclasses of
PartiallyComparable
.Example
Let the integers ordered by the relation \(a \leq b\): is \(a\) a divisor of \(b\)?
implemented by the
Integer
class:>>> from galactic.examples.arithmetic import Integer >>> Integer(5) <= Integer(10) # 5 is a divisor of 10 True >>> Integer(5) <= Integer(11) # 5 is not a divisor of 11 False >>> Integer(5) >= Integer(11) # 5 is not a multiple of 11 False >>>
- __ge__(other: Any) bool
Test if this element is greater than or equal to the other.
- Parameters:
other – the other element
- Returns:
True
– if this element is greater than or equal to the other.NotImplemented
– if the operation is not implemented between the two objects
- abstract __gt__(other: Any) bool
Test if this element is greater than the other.
- Parameters:
other – the other element
- Returns:
True
– if this element is greater than the other.NotImplemented
– if the operation is not implemented between the two objects
- __le__(other: Any) bool
Test if this element is lesser than or equal to the other.
- Parameters:
other – the other element
- Returns:
True
– if this element is lesser than or equal to the other.NotImplemented
– if the operation is not implemented between the two objects
- abstract __lt__(other: Any) bool
Test if this element is lesser than the other.
- Parameters:
other – the other element
- Returns:
True
– if this element is lesser than the other.NotImplemented
– if the operation is not implemented between the two objects
- bottom(iterable: Iterable[galactic.algebras.poset._element._P]) Iterator[galactic.algebras.poset._element._P]
Compute an iterator over the bottom elements of the iterable.
This operation computes in \(O(n^2)\) where \(n\) is the number of elements in the iterable.
- Parameters:
iterable (
Iterable[_P]
) – An iterable of partially ordered elements.- Returns:
An iterator over the bottom elements.
- Return type:
Example
>>> from galactic.algebras.poset import top >>> values = [{1, 2, 3}, {1, 2, 3, 4}, {2, 3, 4, 5}] >>> list(bottom(list(map(frozenset, values)))) [frozenset({1, 2, 3}), frozenset({2, 3, 4, 5})]
- top(iterable: Iterable[galactic.algebras.poset._element._P]) Iterator[galactic.algebras.poset._element._P]
Compute an iterator over the top elements of the iterable.
This operation computes in \(O(n^2)\) where \(n\) is the number of elements in the iterable.
- Parameters:
iterable (
Iterable[_P]
) – An iterable of partially ordered elements.- Returns:
An iterator over the top elements.
- Return type:
Example
>>> from galactic.algebras.poset import top >>> values = [{1, 2, 3}, {1, 2, 3, 4}, {2, 3, 4, 5}] >>> list(top(list(map(frozenset, values)))) [frozenset({1, 2, 3, 4}), frozenset({2, 3, 4, 5})]
- lower_limit(iterable: Iterable[galactic.algebras.poset._element._P], limit: galactic.algebras.poset._element._P, strict: bool = False) Iterator[galactic.algebras.poset._element._P]
Compute an iterator over the elements greater than the limit.
This operation computes in \(O(n)\) where \(n\) is the number of elements in the iterable.
- Parameters:
iterable (
Iterable[_P]
) – An iterable of partially ordered elements.limit – The lower limit
strict (bool) – Is the comparison strict?
- Returns:
An iterator over the selected elements.
- Return type:
Example
>>> from galactic.algebras.poset import lower_limit >>> values = [{1, 2}, {1, 2, 3}, {2, 3}] >>> list(lower_limit(list(map(frozenset, values)), frozenset({2, 3}))) [frozenset({1, 2, 3}), frozenset({2, 3})]
- upper_limit(iterable: Iterable[galactic.algebras.poset._element._P], limit: galactic.algebras.poset._element._P, strict: bool = False) Iterator[galactic.algebras.poset._element._P]
Compute an iterator over the elements lesser than the limit.
This operation computes in \(O(n)\) where \(n\) is the number of elements in the iterable.
- Parameters:
iterable (
Iterable[_P]
) – An iterable of partially ordered elements.limit – The upper limit
strict (bool) – Is the comparison strict?
- Returns:
An iterator over the selected elements.
- Return type:
Example
>>> from galactic.algebras.poset import upper_limit >>> values = [{1, 2}, {1, 2, 3}, {2, 3}] >>> list(upper_limit(list(map(frozenset, values)), frozenset({2, 3}))) [frozenset({2, 3})]
- class ReversibleCoveringRelation(domain: Optional[Iterable[galactic.algebras.poset._cover._P]] = None)
The
ReversibleCoveringRelation
class.It implements the
AbstractReversibleCoveringRelation
class.- __copy__() galactic.algebras.poset.ReversibleCoveringRelation[galactic.algebras.poset._cover._P]
Get a copy of the covering relation.
Its complexity is in \(O(n)\).
- Returns:
The copy of this covering relation.
- Return type:
- __init__(domain: Optional[Iterable[galactic.algebras.poset._cover._P]] = None)
Initialise a
ReversibleCoveringRelation
instance.The initializer can take 0 or 1 argument that should be iterable.
- property co_domain: MutableSet[galactic.algebras.poset._cover._P]
Get the co-domain of this covering relation.
- Returns:
The co-domain of this covering relation.
- Return type:
Notes
The domain and the co-domain of a covering relation are identical.
- property domain: MutableSet[galactic.algebras.poset._cover._P]
Get the domain of this covering relation.
- Returns:
The domain of this covering relation.
- Return type:
- extend(iterable: Iterable[galactic.algebras.poset._cover._P]) None
Extend this covering relation.
- Parameters:
iterable (
Iterable[_P]
) – An iterable of values.
- isdisjoint(other)
Return True if two sets have a null intersection.
- neighbours() Reversible[Tuple[galactic.algebras.poset._cover._P, AbstractSet[galactic.algebras.poset._cover._P], AbstractSet[galactic.algebras.poset._cover._P]]]
Get the neighbourhood of elements.
- Returns:
- Return type:
- predecessors(element: galactic.algebras.poset._cover._P) AbstractSet[galactic.algebras.poset._cover._P]
Get the predecessors of an
element
.- Parameters:
element (
_P
) – The element whose predecessors are requested- Returns:
The predecessors.
- Return type:
- Raises:
ValueError – If the element does not belong to the covering relation.
- property sinks: AbstractSet[galactic.algebras.poset._cover._P]
Get the elements with no successors.
- Returns:
The elements with no successors.
- Return type:
- property sources: AbstractSet[galactic.algebras.poset._cover._P]
Get the elements with no predecessors.
- Returns:
The elements with no predecessors.
- Return type:
- successors(element: galactic.algebras.poset._cover._P) AbstractSet[galactic.algebras.poset._cover._P]
Get the successors of an
element
.- Parameters:
element (
_P
) – The element whose successors are requested- Returns:
The successors.
- Return type:
- Raises:
ValueError – If the element does not belong to the covering relation.
- property universes: Tuple[MutableSet[galactic.algebras.poset._cover._P], MutableSet[galactic.algebras.poset._cover._P]]
Get the universes of this covering relation.
- Returns:
The universes of this covering relation.
- Return type:
- class PartiallyOrderedSet(domain: Optional[Iterable[galactic.algebras.poset._posets._P]] = None)
The
PartiallyOrderedSet
implements posets.A
PartiallyOrderedSet
instance stores its values in two dictionaries for the immediate successors (\(\succ\)) and the immediate predecessors (\(\prec\)).Its memory complexity is in \(O(n)\).
Operation complexities:
__contains__()
: \(O(1)\)__len__()
: \(O(1)\)__iter__()
: \(O(1)\)top()
: \(O(1)\)bottom()
: \(O(1)\)successors()
: \(O(1)\)predecessors()
: \(O(1)\)
Example
>>> from galactic.algebras.poset import PartiallyOrderedSet >>> from galactic.examples.arithmetic import Integer >>> poset = PartiallyOrderedSet[Integer]( ... domain=[ ... Integer(2*3*5*7), ... Integer(3*5*7*11), ... Integer(3*5*7), ... Integer(3*5), ... Integer(5), ... Integer(7), ... ] ... ) >>>
It’s possible to iterate over the poset elements. The elements are iterated level by level starting from the top ones.
Example
>>> sorted(list(map(int, poset.domain))) [5, 7, 15, 105, 210, 1155]
It’s possible to know the poset length.
Example
>>> len(poset.domain) 6
It’s possible to know if an element is in the poset.
Example
>>> Integer(210) in poset.domain True >>> Integer(211) in poset.domain False
It’s possible to iterate over the top and bottom elements.
Example
>>> sorted(list(map(int, poset.top))) [210, 1155] >>> sorted(list(map(int, poset.bottom))) [5, 7]
It’s possible to iterate over the descendants and the ascendants of an element.
Example
>>> sorted(list(map(int, poset.filter(Integer(105)).domain))) [105, 210, 1155] >>> sorted(list(map(int, poset.ideal(Integer(105)).domain))) [5, 7, 15, 105]
It’s possible to iterate over the (immediate) successors and the (immediate) predecessors of an element.
Example
>>> sorted(list(map(int, poset.cover.successors(Integer(105))))) [210, 1155] >>> sorted(list(map(int, poset.successors(Integer(105))))) [105, 210, 1155] >>> sorted(list(map(int, poset.cover.predecessors(Integer(105))))) [7, 15] >>> sorted(list(map(int, poset.predecessors(Integer(105))))) [5, 7, 15, 105]
It’s possible to enlarge a partially ordered set.
Example
>>> poset.extend([Integer(13)]) >>> sorted(list(map(int, poset.domain))) [5, 7, 13, 15, 105, 210, 1155]
- __copy__() galactic.algebras.poset.PartiallyOrderedSet[galactic.algebras.poset._posets._P]
Get a copy of the partially ordered set.
Its complexity is in \(O(n)\).
- Returns:
The copy of this poset.
- Return type:
- __init__(domain: Optional[Iterable[galactic.algebras.poset._posets._P]] = None)
Initialise a
PartiallyOrderedSet
instance.The initializer can take 0 or 1 argument that should be iterable.
- __iter__() Iterator[Tuple[galactic.algebras.poset._abstract._P, galactic.algebras.poset._abstract._P]]
Get an iterator over the elements.
- Returns:
An iterator over the elements
- Return type:
- property bottom: AbstractSet[galactic.algebras.poset._posets._P]
Get the bottom elements.
- Returns:
The bottom elements.
- Return type:
- property co_domain: MutableSet[galactic.algebras.poset._posets._P]
Get the co-domain of this poset.
- Returns:
The co-domain of this poset.
- Return type:
Notes
The domain and the co-domain of a poset are identical.
- property cover: galactic.algebras.poset.ReversibleCoveringRelation[galactic.algebras.poset._posets._P]
Get the covering relation of this poset.
- Returns:
The covering relation.
- Return type:
- property domain: MutableSet[galactic.algebras.poset._posets._P]
Get the domain of this poset.
- Returns:
The domain of this poset.
- Return type:
- extend(iterable: Iterable[galactic.algebras.poset._posets._P]) None
Extend this poset.
- Parameters:
iterable (
Iterable[_P]
) – An iterable of values.
- filter(element: galactic.algebras.poset._posets._P) galactic.algebras.poset.AbstractLowerBoundedSet[galactic.algebras.poset._posets._P]
Get a filter of a poset.
- Parameters:
element (
_P
) – The lower limit- Returns:
The lower bounded poset.
- Return type:
- Raises:
ValueError – If the element does not belong to the poset.
- ideal(element: galactic.algebras.poset._posets._P) galactic.algebras.poset.AbstractUpperBoundedSet[galactic.algebras.poset._posets._P]
Get an ideal of the poset.
- Parameters:
element (
_P
) – The upper limit- Returns:
The upper bounded set.
- Return type:
- isdisjoint(other)
Return True if two sets have a null intersection.
- predecessors(element: galactic.algebras.poset._posets._P) AbstractSet[galactic.algebras.poset._posets._P]
Get the predecessors of an
element
.- Parameters:
element (
_P
) – The element whose predecessors are requested- Returns:
The predecessors.
- Return type:
- Raises:
ValueError – If the element does not belong to the poset.
- property sinks: AbstractSet[galactic.algebras.poset._posets._P]
Get the sink elements.
- Returns:
The sink elements.
- Return type:
- property sources: AbstractSet[galactic.algebras.poset._posets._P]
Get the source elements.
- Returns:
The source elements.
- Return type:
- successors(element: galactic.algebras.poset._posets._P) AbstractSet[galactic.algebras.poset._posets._P]
Get the successors of an
element
.- Parameters:
element (
_P
) – The element whose successors are requested- Returns:
The successors.
- Return type:
- Raises:
ValueError – If the element does not belong to the poset.
- property top: AbstractSet[galactic.algebras.poset._posets._P]
Get the top elements.
- Returns:
The top elements.
- Return type:
- property universes: Tuple[MutableSet[galactic.algebras.poset._posets._P], MutableSet[galactic.algebras.poset._posets._P]]
Get the universes of this poset.
- Returns:
The universes of this poset.
- Return type:
- class DynamicDiagram(domain_renderer: Optional[galactic.algebras.relational.NodeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]] = None, edge_renderer: Optional[galactic.algebras.relational.EdgeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]] = None)
The
DynamicDiagram
class is designed to allow dynamic graph rendering.- __init__(domain_renderer: Optional[galactic.algebras.relational.NodeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]] = None, edge_renderer: Optional[galactic.algebras.relational.EdgeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]] = None)
Initialise a
DynamicDiagram
instance.- Parameters:
domain_renderer (
NodeRenderer[_P, _P]
) – A node renderer.edge_renderer (
EdgeRenderer[_P, _P]
) – An edge renderer.
- property body: Sequence[str]
Get the current graphviz body.
- Returns:
The current body.
- Return type:
- render(filename: str) None
Render the diagram in a file.
- Parameters:
filename (str) – A file path.
- property source: str
Get the graphviz source for this diagram.
- Returns:
The graphviz output for this diagram.
- Return type:
- step(element: galactic.algebras.poset._renderer._P, successors: AbstractSet[galactic.algebras.poset._renderer._P], predecessors: AbstractSet[galactic.algebras.poset._renderer._P]) None
Insert a new element in the diagram.
- Parameters:
element (
_P
) – The new elementsuccessors (
AbstractSet[_P]
) – The immediate successors of the element.predecessors (
AbstractSet[_P]
) – The immediate predecessors of the element.
- class HasseDiagram(poset: galactic.algebras.poset.AbstractPartiallyOrderedSet[galactic.algebras.poset._renderer._P], domain_renderer: Optional[galactic.algebras.relational.NodeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]] = None, edge_renderer: Optional[galactic.algebras.relational.EdgeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]] = None)
The
HasseDiagram
class is used for drawing Hasse diagrams.- __init__(poset: galactic.algebras.poset.AbstractPartiallyOrderedSet[galactic.algebras.poset._renderer._P], domain_renderer: Optional[galactic.algebras.relational.NodeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]] = None, edge_renderer: Optional[galactic.algebras.relational.EdgeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]] = None) None
Initialise a Hasse diagram instance.
- Parameters:
poset (
AbstractPartiallyOrderedSet[_P]
) – A poset.domain_renderer (
NodeRenderer[_P, _P]
) – A node renderer.edge_renderer (
EdgeRenderer[_P, _P]
) – An edge renderer.
- property domain_renderer: galactic.algebras.relational.NodeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]
Get the domain renderer.
- Returns:
The domain renderer.
- Return type:
- property edge_renderer: galactic.algebras.relational.EdgeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]
Get the edge renderer.
- Returns:
The edge renderer.
- Return type:
- property poset: galactic.algebras.poset.AbstractPartiallyOrderedSet[galactic.algebras.poset._renderer._P]
Get the poset.
- Returns:
The underlying poset.
- Return type:
- class IterativeDiagram(poset: galactic.algebras.poset.AbstractPartiallyOrderedSet[galactic.algebras.poset._renderer._P], domain_renderer: Optional[galactic.algebras.relational.NodeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]] = None, edge_renderer: Optional[galactic.algebras.relational.EdgeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]] = None)
The
IterativeDiagram
class allows iterative rendering.- __init__(poset: galactic.algebras.poset.AbstractPartiallyOrderedSet[galactic.algebras.poset._renderer._P], domain_renderer: Optional[galactic.algebras.relational.NodeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]] = None, edge_renderer: Optional[galactic.algebras.relational.EdgeRenderer[galactic.algebras.poset._renderer._P, galactic.algebras.poset._renderer._P]] = None)
Initialise a
IterativeDiagram
instance.- Parameters:
poset (
AbstractPartiallyOrderedSet[_P]
) – A poset.domain_renderer (
NodeRenderer[_P, _P]
) – A node renderer.edge_renderer (
EdgeRenderer[_P, _P]
) – An edge renderer.