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cyclically reduced
Let $M(X)$ be a free monoid with involution ${}^{{1}}$ on $X$. A word $w\in M(X)$ is said to be cyclically reduced if every cyclic conjugate of it is reduced. In other words, $w$ is cyclically reduced iff $w$ is a reduced word and that if $w=uvu^{{1}}$ for some words $u$ and $v$, then $w=v$.
For example, if $X=\{a,b,c\}$, then words such as
$c^{{1}}bc^{2}a\qquad\mbox{and}\qquad abac^{2}ba^{2}$ 
are cyclically reduced, where as words
$a^{2}bca^{{1}}\qquad\mbox{and}\qquad cb^{2}b^{3}c$ 
are not, the former is reduced, but of the form $a(abc)a^{{1}}$, while the later is not even a reduced word.
Remarks. The concept of cyclically reduced words carries over to words in groups. We consider words in a group $G$.

If a word is cyclically reduced, so is its inverse and all of its cyclic conjugates.

A word $v$ is a cyclic reduction of a word $w$ if $w=uvu^{{1}}$ for some word $u$, and $v$ is cyclically reduced. Clearly, every word and its cyclic reduction are conjugates of each other. Furthermore, any word has a unique cyclic reduction.

Every group $G$ has a presentation $\langle SR\rangle$ such that
(a) $R$ is cyclically reduced (meaning every element of $R$ is cyclically reduced),
(b) closed under inverses (meaning if $u\in R$, then $u^{{1}}\in R$), and
(c) closed under cyclic conjugation (meaning any cyclic conjugate of an element in $R$ is in $R$).
Furthermore, if $G$ is finitely presented, $R$ above can be chosen to be finite.
Proof.
Every group $G$ has a presentation $\langle SR^{{\prime}}\rangle$. There is an isomorphism from $F(S)/N(R^{{\prime}})$ to $G$, where $F(S)$ is the free group freely generated by $S$, and $N(R^{{\prime}})$ is the normalizer of the subset $R^{{\prime}}\subseteq F(S)$ in $F(S)$. Let $R^{{\prime\prime}}$ be the set of all cyclic reductions of words in $R^{{\prime}}$. Then $N(R^{{\prime\prime}})=N(R^{{\prime}})$, since any word not cyclically reduced in $R^{{\prime}}$ is conjugate to its cyclic reduction in $R^{{\prime\prime}}$, and hence in $N(R^{{\prime\prime}})$. Next, for each $u\in R^{{\prime\prime}}$, toss in its inverse and all of its cyclic conjugates. The resulting set $R$ is still cyclically reduced. Furthermore, $R$ satisfies the remaining conditions above. Finally, $N(R)=N(R^{{\prime\prime}})$, as any cyclic conjugate $v$ of a word $w$ is clearly a conjugate of $w$. Therefore, $G$ has presentation $\langle SR\rangle$.
If $G$ is finitely presented, then $S$ and $R^{{\prime}}$ above can be chosen to be finite sets. Therefore, $R^{{\prime\prime}}$ and $R$ are both finite. $R$ is finite because the number of cyclic conjugates of a word is at most the length of the word, and hence finite. ∎
Mathematics Subject Classification
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