Elementary Operators for Words#

Multiplication and Exponentiation#

The word operations defined here may be applied either to the words of a free semigroup or the words of a semigroup with non-trivial relations.

u * v: SgpFPElt, SgpFPElt -> SgpFPElt#: Given words \(u\) and \(v\) belonging to the same fp-semigroup \(S\), return the product of \(u\) and \(v\).

u ^ n: SgpFPElt, RngIntElt -> SgpFPElt#: The \(n\)-th power of the word \(u\), where \(n\) is a positive integer.

G ! Q: SgpFP, [ SgpFPElt ] -> SgpFPElt#: Given a sequence \(Q\) of words belonging to the fp-semigroup \(G\), return the product \(Q[1] Q[2] \cdots Q[n]\) of the terms of \(Q\) as a word in \(G\).

The Length of a Word#

# u: SgpFPElt -> RngIntElt#: The length of the word \(u\).

Equality and Comparison#

The words of an fp-semigroup \(S\) are ordered first by length and then lexicographically. The lexicographic ordering is determined by the following ordering on the generators:

\[S.1 < S.2 < S.3 < S.4 < \cdots\]

Here, \(u\) and \(v\) are words belonging to some common fp-semigroup.

u eq v: SgpFPElt, SgpFPElt -> BoolElt#: Returns true if the words \(u\) and \(v\) are identical (as elements of the appropriate free semigroup), false otherwise.

u ne v: SgpFPElt, SgpFPElt -> BoolElt#: Returns true if the words \(u\) and \(v\) are not identical (as elements of the appropriate free semigroup), false otherwise.

u lt v: SgpFPElt, SgpFPElt -> BoolElt#: Returns true if the word \(u\) precedes the word \(v\), with respect to the ordering defined above for elements of an fp-semigroup, false otherwise.

u le v: SgpFPElt, SgpFPElt -> BoolElt#: Returns true if the word \(u\) either precedes, or is equal to, the word \(v\), with respect to the ordering defined above for elements of an fp-semigroup, false otherwise.

u ge v: SgpFPElt, SgpFPElt -> BoolElt#: Returns true if the word \(u\) either follows, or is equal to, the word \(v\), with respect to the ordering defined above for elements of an fp-semigroup, false otherwise.

u gt v: SgpFPElt, SgpFPElt -> BoolElt#: Returns true if the word \(u\) follows the word \(v\), with respect to the ordering defined above for elements of an fp-semigroup.

IsOne(u): MonFPElt -> BoolElt#: Returns true if the word \(u\), belonging to the monoid \(M\), is the identity word, false otherwise.