Elementary Tietze Transformations

AddRelation(S, r): SgpFP, Rel -> SgpFP

AddRelation(S, r, i): SgpFP, Rel, RngIntElt -> SgpFP

Given an fp-semigroup \(S\) and a relation \(r\) in the generators of \(S\), create the quotient semigroup obtained by adding the relation \(r\) to the defining relations of \(S\). If an integer \(i\) is specified as third argument, insert the new relation after the \(i\)-th relation of \(S\). If the third argument is omitted, \(r\) is added to the end of the relations that are carried across from \(S\).

DeleteRelation(S, r): SgpFP, Rel -> SgpFP

Given an fp-semigroup \(S\) and a relation \(r\) that occurs among the given defining relations for \(S\), create the semigroup \(T\), having the same generating set as \(S\) but with the relation \(r\) removed.

DeleteRelation(S, i): SgpFP, RngIntElt -> SgpFP

Given an fp-semigroup \(S\) and an integer \(i\), \(1 \le i \le m\), where \(m\) is the number of defining relations for \(S\), create the semigroup \(T\) having the same generating set as \(S\) but with the \(i\)-th relation omitted.

ReplaceRelation(S, r₁, r₂): SgpFP, Rel, Rel -> SgpFP

Given an fp-semigroup \(S\) and relations \(r_1\) and \(r_2\) in the generators of \(S\), where \(r_1\) is one of the given defining relations for \(S\), create the semigroup \(T\) having the same generating set as \(S\) but with the relation \(r_1\) replaced by the relation \(r_2\).

ReplaceRelation(S, i, r): SgpFP, RngIntElt, Rel -> SgpFP

Given an fp-semigroup \(S\), an integer \(i\), \(1 \le i \le m\), where \(m\) is the number of defining relations for \(S\), and a relation \(r\) in the generators of \(S\), create the semigroup \(T\) having the same generating set as \(S\) but with the \(i\)-th relation of \(S\) replaced by the relation \(r\).

AddGenerator(S): SgpFP -> SgpFP

Given an fp-semigroup \(S\) with presentation \(< X\ |\ R >\), create the semigroup \(T\) with presentation \(< X \cup \{ y\rbrace\ |\ R >\), where \(y\) denotes a new generator.

AddGenerator(S, w): SgpFP, SgpFPElt -> SgpFP

Given an fp-semigroup \(S\) with presentation \(< X\ |\ R >\) and a word \(w\) in the generators of \(S\), create the semigroup \(T\) with presentation \(< X \cup \{ y \rbrace\ |\ R \cup \{ y = w\rbrace >\), where \(y\) denotes a new generator.

DeleteGenerator(S, y): SgpFP, SgpFPElt -> SgpFP

Given an fp-semigroup \(S\) with presentation \(< X\ |\ R >\) and a generator \(y\) of \(S\) such that either \(S\) has no relations involving \(y\), or a single relation \(r\) containing a single occurrence of \(y\), create the semigroup \(T\) with presentation \(< X - \{ y \rbrace\ |\ R - \{ r \rbrace >\).