Converting to a ToddCoxeter

This page contains documentation relating to converting libsemigroups_pybind11 objects into ToddCoxeter instances using the to function.

See also

The to function for an overview of possible conversions between libsemigroups_pybind11 types.

Various uses

Recall that the signature for the to function is to(*args, Return). In what follows, we explain how different values of args and Return may be used to construct ToddCoxeter objects. The following options are possible:

• Converting a FroidurePin to a ToddCoxeter; and

• Converting a KnuthBendix to a ToddCoxeter.

Converting a FroidurePin to a ToddCoxeter

To construct a ToddCoxeter from a FroidurePin, specify all of the following values for args:

• knd (congruence_kind) – the kind of the congruence being construed;

• fpb (FroidurePin) – the FroidurePin instance to be converted; and

• wg (WordGraph) – the left or right Cayley graph of fpb.

Additionally, specify one of the following for Return:

• (ToddCoxeter, str) for constructing a ToddCoxeter on words with type str.

• (ToddCoxeter, list[int]) for constructing a ToddCoxeter on words with type list[int].

This function converts the FroidurePin object fpb into a ToddCoxeter object using the WordGraph wg (which should be either the FroidurePin.left_cayley_graph or the FroidurePin.right_cayley_graph of fpb).

This returned ToddCoxeter object represents the trivial congruence over the semigroup defined by fpb.

This will throw a LibsemigroupsError if wg is not the FroidurePin.left_cayley_graph or the FroidurePin.right_cayley_graph of fpb.

>>> from libsemigroups_pybind11 import (
...     Bipartition,
...     congruence_kind,
...     FroidurePin,
...     to,
...     ToddCoxeter,
... )

>>> b1 = Bipartition([[1, -1], [2, -2], [3, -3], [4, -4]])
>>> b2 = Bipartition([[1, -2], [2, -3], [3, -4], [4, -1]])
>>> b3 = Bipartition([[1, -2], [2, -1], [3, -3], [4, -4]])
>>> b4 = Bipartition([[1, 2], [3, -3], [4, -4], [-1, -2]])
>>> S = FroidurePin(b1, b2, b3, b4)

>>> tc = to(
...     congruence_kind.twosided,   # knd
...     S,                          # fpb
...     S.right_cayley_graph(),     # wg
...     rtype=(ToddCoxeter, str),
... )

>>> tc.run()
>>> S.size() == tc.number_of_classes()
True

Converting a KnuthBendix to a ToddCoxeter

To construct a ToddCoxeter from a KnuthBendix specify all of the following values for args:

• knd (congruence_kind) – the kind of the congruence being constructed.

• kb (KnuthBendix) – the KnuthBendix object being converted.

Additionally, specify the following for Return:

• (ToddCoxeter,) for constructing a ToddCoxeter.

This function converts the KnuthBendix object kb into a ToddCoxeter object using the right Cayley graph of the semigroup represented by kb.

This returned ToddCoxeter object represents the trivial congruence over the semigroup defined by kb.

This will throw a LibsemigroupsError if either:

• kb.kind() is not congruence_kind.twosided; or

• kb.number_of_classes() is not finite. In this case, use ToddCoxeter(knd, kb.presentation()) instead.

>>> from libsemigroups_pybind11 import (
...     congruence_kind,
...     to,
...     KnuthBendix,
...     Presentation,
...     presentation,
...     ToddCoxeter,
... )

>>> p = Presentation('ab')
>>> presentation.add_rule(p, 'ab', 'ba')
>>> presentation.add_rule(p, 'aa', 'a')
>>> presentation.add_rule(p, 'bb', 'b')

>>> kb = KnuthBendix(congruence_kind.twosided, p)

>>> tc = to(
...     congruence_kind.twosided,   # knd
...     kb,                         # kb
...     rtype=(ToddCoxeter,)
... )
>>> tc.run()

>>> tc.number_of_classes() == kb.number_of_classes()
True