Specification of a Presentation

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, ..., x_r, and a set of relations relations over these generators, first construct the free semigroup F on the generators x₁, ..., 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:

(a)

The quotient semigroup S;

(b)

The natural homomorphism φ : F -> S.

Thus, the statement

S< y₁, ..., y_r > := Semigroup< x₁, ..., x_r | w₁, ..., w_s >;

is an abbreviation for

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

S< y₁, ..., y_r > := quo< F | w₁, ..., w_s >;

Monoid< generators | relations > : MonFPElt, ..., MonFPElt, Rel, ..., Rel -> MonFP

Given a generators clause consisting of a list of variables x₁, ..., x_r, and a set of relations relations over these generators, first construct the free monoid F on the generators x₁, ..., 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:

(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, ..., x_r > := FreeMonoid(r);

M< y₁, ..., y_r > := quo< F | w₁, ..., 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)