#5648 ⟨
a
,
b
|
aabaab
=
aaabb
⟩
Up:
Monoids with two generators and one relation
Prev:
#5647
⟨
a
,
b
|
aabaab
=
aaaba
⟩
Next:
#5649
⟨
a
,
b
|
aabaab
=
aabaa
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 11
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
ab
a
2
b
=
a
3
b
2
and
a
⋅
a
2
b
2
=
a
3
b
2
, however
ab
a
2
b
≠
a
2
b
2
Not right cancellative, because right multiplication by
b
is not injective:
a
2
b
a
2
⋅
b
=
a
3
b
2
and
a
3
b
⋅
b
=
a
3
b
2
, however
a
2
b
a
2
≠
a
3
b
Enveloping group: ⟨
a
,
b
|
aaba
-1
b
-1
⟩
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 recursive path with deg(
a
) = 0; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
3
b
2
⇒ (
a
2
b
)
2
[1]
# ab:aabaab=aaabb a/b aaabb=aabaab