Subsemigroups, Ideals and Quotients#

Subsemigroups and Ideals#

sub<S | L₁, ..., Lᵣ>: 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 \(L_1\),\(\ldots\), \(L_r\).

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, ..., Lᵣ>: 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 \(L_1\),\(\ldots\), \(L_r\). The possible forms of a term \(L_i\) of the generator list are the same as for the sub-constructor.

lideal<G | L₁, ..., Lᵣ>: 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 \(L_1\),\(\ldots\), \(L_r\). The possible forms of a term \(L_i\) of the generator list are the same as for the sub-constructor.

rideal<G | L₁, ..., Lᵣ>: 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 \(L_1\),\(\ldots\), \(L_r\). 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 over the generators of \(F\), construct the quotient of \(F\) by the ideal of \(F\) defined by . 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 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 ‘=’: \(w_1 = w_2 = \cdots = w_r\). This is interpreted as the relations \(w_1 = w_r, \ldots, w_{r - 1} = w_r\).

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 \(\phi : F \rightarrow S\).