Specification of a Presentation

Contents

Relations

w1 = w2 : SgpFPElt, SgpFPElt -> Rel
Given words w1 and w2 over the generators of an fp-semigroup S, create the relation w1 = w2. 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 generators clause consisting of a list of variables x1, ..., xr, and a set of relations relations over these generators, first construct the free semigroup F on the generators x1, ..., xr and then construct the quotient of F corresponding to the ideal of F defined by relations.

The syntax for the relations clause is the same as for the quo-constructor. The function returns:

(a)
The quotient semigroup S;
(b)
The natural homomorphism φ : F -> 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 generators clause consisting of a list of variables x1, ..., xr, and a set of relations relations over these generators, first construct the free monoid F on the generators x1, ..., xr and then construct the quotient of F corresponding to the ideal of F defined by relations.

The syntax for the relations clause is the same as for the quo-constructor. The function returns:

(a)
The quotient monoid M;
(b)
The natural homomorphism φ : F -> 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 SgpFP_Monoid (H84E2)

We create the monoid defined by the presentation < x, y | x2, y2, (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)

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.
V2.28, 13 July 2023