Subsections

• Relations
• Presentations
• Accessing the Defining Generators and Relations

w_1 = w_2 : 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).

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.

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.

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 x_1, ..., x_r, and a set of relations relations over these generators, first construct the free semigroup F on the generators x_1, ..., x_r 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:

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

Thus, the statement

S< y_1, ..., y_r > := Semigroup< x_1, ..., x_r | w_1, ..., w_s >;

is an abbreviation for

F< x_1, ..., x_r > := FreeSemigroup(r);

S< y_1, ..., y_r > := quo< F | w_1, ..., w_s >;

Given a generators clause consisting of a list of variables x_1, ..., x_r, and a set of relations relations over these generators, first construct the free monoid F on the generators x_1, ..., x_r 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:

• The quotient monoid M;
• The natural homomorphism phi : F -> M.

Thus, the statement

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

is an abbreviation for

F< x_1, ..., x_r > := FreeMonoid(r);

M< y_1, ..., y_r > := quo< F | w_1, ..., w_s >;

> 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)

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.

A set containing the generators for S.

Ngens(S) : SgpFP -> RngIntElt

The number of generators for S.

The parent semigroup S of the word u.

A sequence containing the defining relations for S.