Subsections

• Subsemigroups and Ideals
• Quotients

Construct the subsemigroup R of the fp-semigroup S generated by the words specified by the terms of the generator list L_1, ..., L_r.

A term L_i of the generator list may consist of any of the following objects:

• A word;
• A set or sequence of words;
• A sequence of integers representing a word;
• A set or sequence of sequences of integers representing words;
• A subsemigroup of an fp-semigroup;
• A set or sequence of subsemigroups.

The collection of words and semigroups specified by the list must all belong to the semigroup S, and R will be constructed as a subgroup of S.

The generators of R consist of the words specified directly by terms L_i together with the stored generating words for any semigroups specified by terms of L_i. Repetitions of an element and occurrences of the identity element are removed (unless R is trivial).

Construct the two-sided ideal I of the fp-semigroup S generated by the words specified by the terms of the generator list L_1, ..., L_r.

The possible forms of a term L_i of the generator list are the same as for the sub-constructor.

Construct the left ideal I of the fp-semigroup S generated by the words specified by the terms of the generator list L_1, ..., L_r.

The possible forms of a term L_i of the generator list are the same as for the sub-constructor.

Construct the right ideal I of the fp-semigroup S generated by the words specified by the terms of the generator list L_1, ..., L_r.

The possible forms of a term L_i of the generator list are the same as for the sub-constructor.

Given an fp-semigroup F, and a list of relations relations over the generators of F, construct the quotient of F by the ideal of F defined by relations.

The expression defining F may be either simply the name of a previously constructed semigroup, or an expression defining an fp-semigroup.

Each term of the list relations must be a relation, a relation list or, if S is a monoid, a word.

A word is interpreted as a relator if S is a monoid.

A relation consists of a pair of words, separated by `='. (See above).

A relation list consists of a list of words, where each pair of adjacent words is separated by `=': w_1 = w_2 = ... = w_r. This is interpreted as the relations w_1 = w_r, ..., w_(r - 1) = w_r.

Note that the relation list construct is only meaningful in the context of the fp semigroup-constructor.

In the context of the quo-constructor, the identity element (empty word) of a monoid may be represented by the digit 1.

Note that this function returns:

• The quotient semigroup S;
• The natural homomorphism phi : F -> S.

Given an ideal I of the semigroup S, construct the quotient of S by the ideal I. The quotient is formed by taking the presentation for S and including the generating words of I as additional relations.