<s>
In	O
group	O
theory	O
,	O
Matsumoto	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
,	O
proved	O
by	O
,	O
gives	O
conditions	O
for	O
two	O
reduced	O
words	O
of	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
to	O
represent	O
the	O
same	O
element	O
.	O
</s>
<s>
is	O
one	O
of	O
the	O
defining	O
relations	O
of	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Matsumoto	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
implies	O
that	O
there	O
is	O
a	O
natural	O
map	O
(	O
not	O
a	O
group	O
homomorphism	O
)	O
from	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
to	O
the	O
corresponding	O
braid	B-Application
group	I-Application
,	O
taking	O
any	O
element	O
of	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
represented	O
by	O
some	O
reduced	O
word	O
in	O
the	O
generators	O
to	O
the	O
same	O
word	O
in	O
the	O
generators	O
of	O
the	O
braid	B-Application
group	I-Application
.	O
</s>
