Specification of a Word

Suppose A is an fp-algebra for which generators have already been defined. A word is defined inductively as follows:

• A generator is a word;
• The expression (u) is a word, where u is a word;
• The sum u + v of two words is a word;
• The difference u - v of two words is a word;
• The product u * v of the words u and v is a word;
• The power of a word, u^n, where u is a word and n is an integer, is a word;
• The product e * u, where e is an element of the coefficient ring and u is a word, is a word;
• The product u * e, where e is an element of the coefficient ring and u is a word, is a word.

Subsections

• Elementary Operators for Elements
• Boolean Operators for Elements
• Elementary Functions for Elements

A ! e : AlgFP, MonFPElt -> AlgFPElt

If the underlying monoid has an identity algorithm, construct the element e * ( Id)(M), where e is an element of the coefficient ring R and M is the monoid.

Elementary Operators for Elements

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

u + v : AlgFPElt, AlgFPElt -> AlgFPElt

Given words u and v belonging to the same fp-algebra A, return the sum of u and v.

u - v : AlgFPElt, AlgFPElt -> AlgFPElt

Given words u and v belonging to the same fp-algebra A, return the difference of u and v.

u * v : AlgFPElt, AlgFPElt -> AlgFPElt

Given words u and v belonging to the same fp-algebra A, return the product of u and v.

u ^ n : AlgFPElt, RngIntElt -> AlgFPElt

The n-th power of the word u, where n is a positive integer.

Boolean Operators for Elements

The words of an fp-algebra A are ordered first by length and then lexicographically. The lexicographic ordering is determined by the ordering on the coefficient ring and the monoid. Here, u and v are words belonging to some common fp-algebra.

u eq v : AlgFPElt, AlgFPElt -> BoolElt

True if the words u and v are identical (as elements of the appropriate free algebra), false otherwise.

u ne v : AlgFPElt, AlgFPElt -> BoolElt

Returns true if the words u and v are not identical (as elements of the appropriate free algebra), false otherwise.

u lt v : AlgFPElt, AlgFPElt -> BoolElt

Returns true if the word u precedes the word v, with respect to the ordering defined above for elements of an fp-algebra, false otherwise.

u le v : AlgFPElt, AlgFPElt -> 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-algebra, false otherwise.

u ge v : AlgFPElt, AlgFPElt -> 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-algebra, false otherwise.

u gt v : AlgFPElt, AlgFPElt -> BoolElt

True if the word u follows the word v, with respect to the ordering defined above for elements of an fp-algebra.

IsZero(u) : AlgFPElt -> BoolElt

Returns true if u is zero (the empty word), false otherwise.

IsScalar(u) : AlgFPElt -> BoolElt

Returns true if u is a scalar, that is, an element of the underlying ring, false otherwise.

Elementary Functions for Elements

# u : AlgFPElt -> RngIntElt

The length of the word u.

Support(u) : AlgFPElt -> [ MonElt ]

Returns the sequence of words whose coefficients in u are non-zero.

LeadingCoefficient(u) : AlgFPElt -> RngElt

Returns the coefficient of the highest occurring power of the most principal monoid generator (where M.i is more principal than M.j if and only if i>j).

MonomialCoefficient(u, m) : AlgFPElt, MonElt -> RngElt

Returns the coefficient of the monoid element m in u.