sign_vectors.functions¶

Functions for sets of sign vectors¶

Functions

`contraction`(iterable, indices)	Return all sign vectors or vectors that are zero on given components.
`deletion`(iterable, indices)	Remove given components from an iterable of sign vectors
`lower_closure`(iterable)	Compute the lower closure of given sign vectors.
`orthogonal_complement`(iterable)	Compute the orthogonal complement of given sign vectors.
`plot_sign_vectors`(iterable[, vertex_size, ...])	Plot the Hasse Diagram of sign vectors using the conformal relation.
`total_closure`(iterable)	Compute the total closure of given sign vectors.
`upper_closure`(iterable)	Compute the upper closure of given sign vectors.

sign_vectors.functions.contraction(iterable: set[SignVector], indices: list[int]) → set[SignVector]¶

Return all sign vectors or vectors that are zero on given components.

INPUT:

iterable – an iterable of sign vectors or vectors
indices – a list of indices.

OUTPUT:

If keep_components is false, remove entries in indices. (default)
If keep_components is true, keep entries in indices.

EXAMPLES:

sage: from sign_vectors import *
sage: W = [sign_vector("++0"), sign_vector("-00"), sign_vector("00+")]
sage: W
[(++0), (-00), (00+)]

We consider the contraction for different indices:

sage: contraction(W, [0])
{(0+)}
sage: contraction(W, [1])
{(-0), (0+)}
sage: contraction(W, [2])
{(-0), (++)}

The second sign vector has zeros at positions 1 and 2:

sage: contraction(W, [1, 2])
{(-)}

sign_vectors.functions.deletion(iterable: set[SignVector], indices: list[int]) → set[SignVector]¶

Remove given components from an iterable of sign vectors

INPUT:

iterable – an iterable of sign vectors
indices – a list of indices

EXAMPLES:

sage: from sign_vectors import *
sage: W = [sign_vector("++0"), sign_vector("00-"), sign_vector("+00")]
sage: W
[(++0), (00-), (+00)]
sage: deletion(W, [0])
{(00), (+0), (0-)}

Duplicate sign vectors are removed if they would occur:

sage: deletion(W, [1])
{(+0), (0-)}
sage: deletion(W, [1, 2])
{(0), (+)}

sign_vectors.functions.lower_closure(iterable: set[SignVector]) → set[SignVector]¶

Compute the lower closure of given sign vectors.

INPUT:

iterable – an iterable of sign vectors

OUTPUT: Return the lower closure of iterable as a set of sign vectors.

Note

The sign vector \(X\) is in the lower closure of a set of sign vectors \(W\) if there exists \(Y \in W\) with \(X \leq Y\).

EXAMPLES:

First, we consider a list consisting of only one sign vector:

sage: from sign_vectors import *
sage: W = [sign_vector("+-0")]
sage: W
[(+-0)]
sage: lower_closure(W)
{(000), (+00), (0-0), (+-0)}

Now, we consider a list of three sign vectors:

sage: W = [sign_vector("++-"), sign_vector("-00"), sign_vector("0--")]
sage: W
[(++-), (-00), (0--)]
sage: lower_closure(W)
{(000), (-00), (00-), (0+0), (0+-), (+00), (+0-), (++0), (++-), (0-0), (0--)}

TESTS:

sage: lower_closure([])
set()

sign_vectors.functions.orthogonal_complement(iterable: set[SignVector]) → set[SignVector]¶

Compute the orthogonal complement of given sign vectors. INPUT:

iterable – an iterable of sign vectors

OUTPUT: Return the orthogonal complement of iterable as a set of partial sign vectors.

EXAMPLES:

We consider a list consisting of only one sign vector:

sage: from sign_vectors import *
sage: W = ExtendedSignVector.from_sign_vector(sign_vector("+-0"))
sage: W
(+-0)
sage: orthogonal_complement([W])
{(000), (---), (00+), (00-), (++0), (++-), (--0), (--+), (+++)}

Now, we consider a list of three sign vectors:

sage: W = [sign_vector("++-"), sign_vector("-00"), sign_vector("0--")]
sage: W
[(++-), (-00), (0--)]
sage: orthogonal_complement(W)
{(000)}
sage: W = [sign_vector("++-0"), sign_vector("+++0"), sign_vector("++00")]
sage: W
[(++-0), (+++0), (++00)]
sage: orthogonal_complement(W)
{(0000),
 (-+-+),
 (+-00),
 (-+00),
 (-++0),
 (000+),
 (-+++),
 (000-),
 (+---),
 (+-0+),
 (-+--),
 (+-+0),
 (+-+-),
 (+-++),
 (+--0),
 (-+-0),
 (+--+),
 (-+0+),
 (-+0-),
 (+-0-),
 (-++-)}

sign_vectors.functions.plot_sign_vectors(iterable: set[SignVector], vertex_size: int = 600, figsize: int | None = None, aspect_ratio=None)¶

Plot the Hasse Diagram of sign vectors using the conformal relation.

INPUT:

iterable – an iterable of sign vectors
vertex_size – the size of the vertices in the plot (default: 600)
figsize – the size of the figure (default: None)
aspect_ratio – the aspect ratio of the plot (default: None)

sign_vectors.functions.total_closure(iterable: set[SignVector]) → set[SignVector]¶

Compute the total closure of given sign vectors.

INPUT:

iterable – an iterable of sign vectors

OUTPUT: Return the total closure of iterable as a set of sign vectors.

Note

The total closure is the union of the lower and upper closure.

See also

lower_closure()
upper_closure()

EXAMPLES:

First, we consider a list consisting of only one sign vector:

sage: from sign_vectors import *
sage: W = [sign_vector("+-0")]
sage: W
[(+-0)]
sage: total_closure(W)
{(000), (0-0), (+00), (+-0), (+--), (+-+)}

Now, we consider a list of three sign vectors:

sage: W = [sign_vector("++-"), sign_vector("-00"), sign_vector("0--")]
sage: W
[(++-), (-00), (0--)]
sage: total_closure(W)
{(000),
 (-00),
 (00-),
 (---),
 (0+-),
 (++-),
 (-+0),
 (0-0),
 (--+),
 (-++),
 (-0-),
 (+00),
 (+--),
 (-+-),
 (-0+),
 (--0),
 (0+0),
 (+0-),
 (++0),
 (0--)}

sign_vectors.functions.upper_closure(iterable: set[SignVector]) → set[SignVector]¶

Compute the upper closure of given sign vectors.

INPUT:

iterable – an iterable of sign vectors

OUTPUT: Return the upper closure of iterable as a set of sign vectors.

Note

The sign vector \(X\) is in the upper closure of a set of sign vectors \(W\) if there exists \(Y \in W\) with \(X \leq Y\).

EXAMPLES:

First, we consider a list consisting of only one sign vector:

sage: from sign_vectors import *
sage: W = [sign_vector("+-0")]
sage: W
[(+-0)]
sage: upper_closure(W)
{(+-0), (+--), (+-+)}

Now, we consider a list of three sign vectors:

sage: W = [sign_vector("++-"), sign_vector("-00"), sign_vector("0--")]
sage: W
[(++-), (-00), (0--)]
sage: upper_closure(W)
{(-00), (---), (++-), (-+0), (--+), (-++), (--0), (0--), (-0-), (+--), (-+-), (-0+)}

TESTS:

sage: upper_closure([])
set()