# 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 $< X - \{ y \rbrace\ |\ R - \{ r \rbrace >$.