#840 ⟨
a
,
b
|
abaabaab
=
b
⟩
Up:
Monoids with two generators and one relation
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#839
⟨
a
,
b
|
abaabaab
=
a
⟩
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#841
⟨
a
,
b
|
abaababa
=
a
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 9
Infinite non-cancellative non-commutative monoid
Not right cancellative, because right multiplication by
b
is not injective:
a
(
b
a
2
)
2
⋅
b
=
b
and 1 ⋅
b
=
b
, however
a
(
b
a
2
)
2
≠ 1
Enveloping group is isomorphic to ℤ
Rewriting system
Format:
Pretty
Plain
Word to reduce:
Tips:
Lowercase letters stand for generators.
Spaces are ignored.
Numbers repeat the previous letter, e.g.
b90
.
Reduction strategy:
Leftmost
Rightmost
Path to normal form:
1
1
Reduction order:
Left-to-right shortlex with
b
<
a
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
b
2
⇒
bab
[3]
2.
(
ab
)
2
⇒
b
a
2
b
[2]
3.
ab
(
a
2
b
)
2
⇒
b
[1]
# ab:abaabaab=b ba abb=bab abab=baab abaabaab=b