Elements¶

This module defines essential type for representing lattice elements.

class galactic.algebras.lattice.elements.Joinable¶

Bases: galactic.algebras.poset.elements.PartiallyOrdered

The Joinable type sets for each pair of elements a, b an unique supremum c = a | b (also called a least upper bound or join).

A class implementing the Joinable abstract class must be declared by inheriting from Joinable with itself as a parameter and must implement the methods:

__lt__()
__gt__()
__eq__()
__or__()

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.algebras import Integer
>>> Integer(24) | Integer(36)
Integer(72)
>>> Integer(24).union(Integer(36), Integer(10))
Integer(360)

abstract __or__(other)¶

Return the join of this element and the other.

Parameters: other – the other element
Returns: the join of this element and the other
Return type: Joinable

union(*args) → galactic.algebras.lattice.elements.Joinable¶

Returns the union of this element and the others.

Parameters: *args (Joinable) – the others elements
Returns: the join of this element and the others
Return type: Joinable
Raises: TypeError – if one of the other elements is not an instance of this element class.

class galactic.algebras.lattice.elements.Meetable¶

Bases: galactic.algebras.poset.elements.PartiallyOrdered

The Meetable type sets for each pair of elements a, b an unique infimum c = a & b (also called a greatest lower bound or meet).

A class implementing the Meetable abstract class must be declared by inheriting from Meetable with itself as a parameter and must implement the methods:

__lt__()
__gt__()
__eq__()
__and__()

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.algebras import Integer
>>> Integer(24) & Integer(36)
Integer(12)
>>> Integer(24).intersection(Integer(36), Integer(10))
Integer(2)

abstract __and__(other)¶

Return the meet of this element and the other.

Parameters: other – the other element
Returns: the meet of this element and the other
Return type: Meetable

intersection(*args) → galactic.algebras.lattice.elements.Meetable¶

Returns the intersection of this element and the others.

Parameters: *args (Meetable) – the others elements
Returns: the meet of this element and the others
Return type: Meetable
Raises: TypeError – if one of the other elements is not an instance of this element class.

class galactic.algebras.lattice.elements.Element¶

Bases: galactic.algebras.lattice.elements.Meetable, galactic.algebras.lattice.elements.Joinable

The Element class describes elements that can be member of a lattice.

galactic.algebras.lattice.elements.infimum(iterable: Iterable[galactic.algebras.lattice.elements.Meetable] = None, default=None) → galactic.algebras.lattice.elements.Meetable¶

Computes the infimum of meetable elements.

Keyword Arguments

iterable (Iterable[Meetable]) – An iterable collection of meetable elements.
default – The default value when the iterable is empty.

Returns

Return type

The intersection of all elements from the iterable collection.

galactic.algebras.lattice.elements.supremum(iterable: Iterable[galactic.algebras.lattice.elements.Joinable] = None, default=None) → galactic.algebras.lattice.elements.Joinable¶

Computes the supremum of joinable elements.

Keyword Arguments

iterable (Iterable[joinable]) – An iterable collection of joinable elements.
default – The default value when the iterable is empty.

Returns

Return type

The union of all elements from the iterable collection.

galactic.algebras.lattice.elements.supremum_generators(iterable: Iterable[galactic.algebras.lattice.elements.Joinable] = None) → Iterator[galactic.algebras.lattice.elements.Joinable]¶

The supremum_generators() function produces the supremum generators from an iterable of joinable elements.

Keyword Arguments: iterable (Iterable[Joinable]) – The iterable collection of elements.
Returns: An iterator over the supremum generators.
Return type: Iterator[Joinable]

galactic.algebras.lattice.elements.infimum_generators(iterable: Iterable[galactic.algebras.lattice.elements.Meetable] = None) → Iterator[galactic.algebras.lattice.elements.Meetable]¶

The infimum_generators() function produces the infimum generators from an iterable of meetable elements.

Keyword Arguments: iterable (Iterable[Meetable]) – The iterable collection of elements.
Returns: An iterator over the infimum generators.
Return type: Iterator[Meetable]