Specification of a Presentation

Relations

w₁ = w₂: SgpFPElt, SgpFPElt -> Rel

Given words \(w_1\) and \(w_2\) over the generators of an fp-semigroup \(S\), create the relation \(w_1 = w_2\). Note that this relation is not automatically added to the existing set of defining relations \(R\) for \(S\). It may be added to \(R\), for example, through use of the quo-constructor (see below).

LHS(r): Rel -> SgpFPElt

Given a relation \(r\) over the generators of \(S\), return the left hand side of the relation \(r\). The object returned is a word over the generators of \(S\).

RHS(r): Rel -> SgpFPElt

Given a relation \(r\) over the generators of \(S\), return the right hand side of the relation \(r\). The object returned is a word over the generators of \(S\).

Presentations

A semigroup with non-trivial relations is constructed as a quotient of an existing semigroup, possibly a free semigroup.

Semigroup< generators | relations >: SgpFPElt, ..., SgpFPElt, Rel, ...Rel -> SgpFP

Given a clause consisting of a list of variables \(x_1, \cdots, x_r\), and a set of relations over these generators, first construct the free semigroup \(F\) on the generators \(x_1, \cdots, x_r\) and then construct the quotient of \(F\) corresponding to the ideal of \(F\) defined by . The syntax for the clause is the same as for the quo-constructor. The function returns:

(a)

The quotient semigroup \(S\);

(b)

The natural homomorphism \(\phi : F \rightarrow S\).

Thus, the statement

    S< y₁, ..., yᵣ > := Semigroup< x₁, ..., xᵣ | w₁, ..., wₛ >;

is an abbreviation for

   F< x₁, ..., xᵣ > := FreeSemigroup(r);

   S< y₁, ..., yᵣ > := quo< F | w₁, ..., wₛ >;
Monoid< generators | relations >: MonFPElt, ..., MonFPElt, Rel, ..., Rel -> MonFP

Given a clause consisting of a list of variables \(x_1, \cdots, x_r\), and a set of relations over these generators, first construct the free monoid \(F\) on the generators \(x_1, \cdots, x_r\) and then construct the quotient of \(F\) corresponding to the ideal of \(F\) defined by . The syntax for the clause is the same as for the quo-constructor. The function returns:

(a)

The quotient monoid \(M\);

(b)

The natural homomorphism \(\phi : F \rightarrow M\).

Thus, the statement

    M< y1, ..., yr > := Monoid< x1, ..., xr | w1, ..., ws >;

is an abbreviation for

   F< x₁, ..., xᵣ > := FreeMonoid(r);

   M< y₁, ..., yᵣ > := quo< F | w₁, ..., wₛ >;
Example: Finitely Presented Monoid (ex-a36e1d)

We create the monoid defined by the presentation \(< x, y\ |\ x^2, y^2, (xy)^2 >\).

> M<x,y> := Monoid< x, y | x^2, y^2, (x*y)^2 >;
> M;
Finitely presented monoid
Relations:
    x^2 = Id(M)
    y^2 = Id(M)
    (x * y)^2 = Id(M)

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Accessing the Defining Generators and Relations

The functions in this group provide access to basic information stored for a finitely-presented semigroup \(G\).

S . i: SgpFP, RngIntElt -> SgpFPElt

The \(i\)-th defining generator for \(S\).

Generators(S): SgpFP -> { SgpFPElt}

A set containing the generators for \(S\).

NumberOfGenerators(S): SgpFP -> RngIntElt
Ngens(S): SgpFP -> RngIntElt

The number of generators for \(S\).

Parent(u): SgpFPElt -> SgpFP

The parent semigroup \(S\) of the word \(u\).

Relations(S): SgpFP -> [ Rel ]

A sequence containing the defining relations for \(S\).