sign_vectors.chirotopes¶

Chirotopes (auxiliary module for oriented matroids)

Chirotopes¶

Classes

`Chirotope`(rank, ground_set_size)	A chirotope of given rank and ground set size.
`Sign`(value)	Class for chirotope entries.

class sign_vectors.chirotopes.Chirotope(rank: int, ground_set_size: int)¶

A chirotope of given rank and ground set size.

EXAMPLES:

sage: from sign_vectors import *
sage: from sign_vectors.chirotopes import *
sage: c = Chirotope.from_list([0, 0, -1, 1, 0, 1, -1, 1, -1, -1], 2, 5)
sage: c
Chirotope of rank 2 on ground set of size 5
sage: c.entry((0, 3))
-
sage: c.entries()
[0, 0, -, +, 0, +, -, +, -, -]
sage: c.as_string()
'00-+0+-+--'
sage: c.oriented_matroid()
Oriented matroid of dimension 1 with elements of size 5.
sage: c.dual()
Chirotope of rank 3 on ground set of size 5
sage: c.dual().entries()
[-, +, +, -, -, 0, -, -, 0, 0]

We construct chirotopes from matrices:

sage: M = matrix([[1, 2, 0, 0], [0, 1, 2, 3]])
sage: c = Chirotope.from_matrix(M)
sage: c.entry((0, 1))
+
sage: c.entries()
[+, +, +, +, +, 0]
sage: c.dual()
Chirotope of rank 2 on ground set of size 4
sage: c.dual().entries()
[0, -, +, +, -, +]

sage: M = matrix([[1, 0, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1]])
sage: c = Chirotope.from_matrix(M)
sage: c
Chirotope of rank 3 on ground set of size 4
sage: c.entries()
[+, +, -, +]
sage: c.dual()
Chirotope of rank 1 on ground set of size 4
sage: c.dual().entries()
[+, +, +, -]

We construct chirotopes from circuits:

sage: c = Chirotope.from_circuits([sign_vector("00+0"), sign_vector("000+")], 2, 4)
sage: c.entry([0, 1])
+
sage: c.entry([0, 2])
0
sage: c.entry([1, 2])
0
sage: c = Chirotope.from_circuits([sign_vector("00+0"), sign_vector("000+")], 2, 4)
sage: c.entries()
[+, 0, 0, 0, 0, 0]
sage: c.dual()
Chirotope of rank 2 on ground set of size 4
sage: c.dual().entries()
[0, 0, 0, 0, 0, +]
sage: c.as_string()
'+00000'

sage: c = Chirotope.from_circuits([sign_vector("++0--"), sign_vector("+0--0"), sign_vector("-0++0"), sign_vector("0--0+"), sign_vector("--0++"), sign_vector("0++0-")], 3, 5)
sage: c.entries()
[+, -, +, 0, -, +, +, 0, -, +]
sage: c.dual().entries()
[+, +, 0, -, +, +, 0, +, +, +]

We construct chirotopes from cocircuits:

sage: c = Chirotope.from_cocircuits([sign_vector("00+0"), sign_vector("000+")], 2, 4)
sage: c.entries()
[0, 0, 0, 0, 0, +]

sage: c = Chirotope.from_cocircuits([sign_vector("00++"), sign_vector("0++0"), sign_vector("0+0-")], 2, 4)
sage: c.entries()
[0, 0, 0, +, +, +]

as_string() → str¶: Represent the chirotope as a string.

dual() → Chirotope¶: Return the dual chirotope.

entries() → list[Sign]¶: Return all chirotope entries in lexicographic order.

entry(rset: list[int]) → Sign¶: Return the chirotope entry given by indices.

static from_circuits(circuits: set[SignVector], rank: int, ground_set_size: int) → Chirotope¶: Construct a chirotope from its circuits.

static from_cocircuits(cocircuits: set[SignVector], rank: int, ground_set_size: int) → Chirotope¶: Construct a chirotope from its cocircuits.

static from_list(entries: list[Sign], rank: int, ground_set_size: int) → Chirotope¶: Construct a chirotope from its entries.

static from_matrix(matrix) → Chirotope¶: Construct a chirotope from a matrix.

oriented_matroid() → OrientedMatroid¶: Return the oriented matroid corresponding to the chirotope.

class sign_vectors.chirotopes.Sign(value)¶

Class for chirotope entries.

EXAMPLES:

sage: from sign_vectors.chirotopes import Sign
sage: Sign(1)
+
sage: Sign(-1)
-
sage: Sign(0)
0
sage: Sign(5)
+
sage: Sign(5).value
1
sage: -Sign(5)
-
sage: Sign("+")
+
sage: Sign("-")
-
sage: Sign("0")
0

as_integer_ratio()¶

Return integer ratio.

Return a pair of integers, whose ratio is exactly equal to the original int and with a positive denominator.

>>> (10).as_integer_ratio()
(10, 1)
>>> (-10).as_integer_ratio()
(-10, 1)
>>> (0).as_integer_ratio()
(0, 1)

bit_count()¶

Number of ones in the binary representation of the absolute value of self.

Also known as the population count.

>>> bin(13)
'0b1101'
>>> (13).bit_count()
3

bit_length()¶

Number of bits necessary to represent self in binary.

>>> bin(37)
'0b100101'
>>> (37).bit_length()
6

conjugate()¶: Returns self, the complex conjugate of any int.

denominator¶: the denominator of a rational number in lowest terms

from_bytes(byteorder, *, signed=False)¶

Return the integer represented by the given array of bytes.

bytes: Holds the array of bytes to convert. The argument must either support the buffer protocol or be an iterable object producing bytes. Bytes and bytearray are examples of built-in objects that support the buffer protocol.
byteorder: The byte order used to represent the integer. If byteorder is ‘big’, the most significant byte is at the beginning of the byte array. If byteorder is ‘little’, the most significant byte is at the end of the byte array. To request the native byte order of the host system, use `sys.byteorder’ as the byte order value.
signed: Indicates whether two’s complement is used to represent the integer.

imag¶: the imaginary part of a complex number

numerator¶: the numerator of a rational number in lowest terms

real¶: the real part of a complex number

to_bytes(length, byteorder, *, signed=False)¶

Return an array of bytes representing an integer.

length: Length of bytes object to use. An OverflowError is raised if the integer is not representable with the given number of bytes.
byteorder: The byte order used to represent the integer. If byteorder is ‘big’, the most significant byte is at the beginning of the byte array. If byteorder is ‘little’, the most significant byte is at the end of the byte array. To request the native byte order of the host system, use `sys.byteorder’ as the byte order value.
signed: Determines whether two’s complement is used to represent the integer. If signed is False and a negative integer is given, an OverflowError is raised.