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Subsemigroups, Ideals and Quotients

Subsemigroups and Ideals

sub<S | L₁, ..., L_r> : SgpFP, SgpFPElt, ..., SgpFPElt -> SgpFP

Construct the subsemigroup R of the fp-semigroup S generated by the words specified by the terms of the generator list L₁, ..., L_r.
A term L_i of the generator list may consist of any of the following objects:

(a)
A word;
(b)
A set or sequence of words;
(c)
A sequence of integers representing a word;
(d)
A set or sequence of sequences of integers representing words;
(e)
A subsemigroup of an fp-semigroup;
(f)
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).

ideal<S | L₁, ..., L_r> : SgpFP, SgpFPElt, ..., SgpFPElt -> SgpFPIdl

Construct the two-sided ideal I of the fp-semigroup S generated by the words specified by the terms of the generator list L₁, ..., L_r.
The possible forms of a term L_i of the generator list are the same as for the sub-constructor.

lideal<G | L₁, ..., L_r> : SgpFP, SgpFPElt, ..., SgpFPElt -> SgpFPIdl

Construct the left ideal I of the fp-semigroup S generated by the words specified by the terms of the generator list L₁, ..., L_r.
The possible forms of a term L_i of the generator list are the same as for the sub-constructor.

rideal<G | L₁, ..., L_r> : SgpFP, SgpFPElt, ..., SgpFPElt -> SgpFPIdl

Construct the right ideal I of the fp-semigroup S generated by the words specified by the terms of the generator list L₁, ..., L_r.
The possible forms of a term L_i of the generator list are the same as for the sub-constructor.

Quotients

quo< F | relations > : SgpFP, Rel, ..., Rel -> SgpFP

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 `=': w₁ = w₂ = ... = w_r. This is interpreted as the relations w₁ = 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:

(a)
The quotient semigroup S;
(b)
The natural homomorphism φ : F -> S.

V2.28, 13 July 2023

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