# Free Bands¶

libsemigroups contains an implementation of the Radoszewski-Rytter algorithm [RR10] for testing equivalence of words in free bands.

The functions in libsemigroups for free bands are described on this page.

bool libsemigroups::freeband_equal_to(word_type const &x, word_type const &y)

Check if two given words represent the same element in the free band.

The free band is the free object in the variety of bands or idempotent semigroups. The free band $$\textrm{FB}(A)$$ generated by some set $$A$$ is the largest semigroup all of whose elements $$x$$ are idempotent, i.e. satisfy $$x^2=x$$. This function efficiently compares whether two words in the generators of $$\textrm{FB}(A)$$ are the same as elements of the free band.

Complexity

The time complexity is $$O(mn)$$ and space complexity is $$O(n)$$ where $$n$$ is the total length of x and y, and $$m$$ is the number of distinct letters appearing in x and y.

Example

using namespace libsemigroups;
freeband_equal_to({0, 1, 2, 3, 2, 1, 0},
{0, 1, 2, 3, 2, 3, 2, 1, 0}); // true
freeband_equal_to({1, 2, 3}, {0, 1, 2}); // false
freeband_equal_to({1, 4, 2, 3, 10}, {1, 4, 1, 4, 2, 3, 10}) // true
freeband_equal_to({0, 1, 2, 3, 4, 0, 1, 2, 3, 4},
{4, 3, 2, 1, 0, 4, 3, 2, 1, 0}); // false
freeband_equal_to({0, 1, 2, 1, 0, 1, 2}, {0, 1, 2}); // true
freeband_equal_to({0, 1, 2, 3, 0, 1},
{0, 1, 2, 3, 3, 2, 2, 1, 0, 2, 1, 0, 2, 3,
0, 2, 1, 3, 2, 1, 2, 3, 2, 1, 0, 2, 0, 1,
0, 2, 0, 3, 2, 0, 1, 2, 2, 3, 0, 1}); // true


Parameters
• x – the first word to compare in the free band.

• y – the second word to compare in the free band.

Throws

(None) – This function guarantees not to throw a LibsemigroupsException.

Returns

Return true if both words are the same as elements of the free band and false otherwise.

template<typename T>
bool libsemigroups::freeband_equal_to(T x, T y)

Check if two given words represent the same element in the free band.

Complexity

The time complexity is $$O(mn)$$ and space complexity is $$O(n)$$ where $$n$$ is the total length of x and y, and $$m$$ is the number of distinct letters appearing in x and y.

Template Parameters

T – any type that can be converted to word_type

Parameters
• x – the first word to compare in the free band.

• y – the second word to compare in the free band.

Throws

(None) – This function guarantees not to throw a LibsemigroupsException.

Returns

Return true if both words are the same as elements of the free band and false otherwise.

template<typename T>
bool libsemigroups::freeband_equal_to(T first1, T last1, T first2, T last2)

Check if two given words represent the same element in the free band.

Complexity

The time complexity is $$O(mn)$$ and space complexity is $$O(n)$$ where $$n$$ is the total length of x and y, and $$m$$ is the number of distinct letters appearing in x and y.

Template Parameters

T – any type that can be converted to word_type

Parameters
• first1 – iterator to start of the first word.

• last1 – iterator to end of the first word.

• first2 – iterator to start of the second word.

• last2 – iterator to end of the second word.

Throws

(None) – This function guarantees not to throw a LibsemigroupsException.

Returns

Return true if both words are the same as elements of the free band and false otherwise.