#2526 ⟨
a
,
b
|
aabbaa
=
abba
⟩
Up:
Monoids with two generators and one relation
Prev:
#2525
⟨
a
,
b
|
aabbaa
=
abab
⟩
Next:
#2527
⟨
a
,
b
|
aabbaa
=
abbb
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 10
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
a
b
2
a
2
=
a
b
2
a
and
a
⋅
b
2
a
=
a
b
2
a
, however
a
b
2
a
2
≠
b
2
a
Not right cancellative, because right multiplication by
a
is not injective:
a
2
b
2
a
⋅
a
=
a
b
2
a
and
a
b
2
⋅
a
=
a
b
2
a
, however
a
2
b
2
a
≠
a
b
2
Enveloping group: ⟨
a
,
b
|
aaaabb
⟩
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
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
2
b
2
a
2
⇒
a
b
2
a
[1]
2.
(
a
b
2
)
2
a
2
⇒
a
2
(
b
2
a
)
2
[2]
# ab:aabbaa=abba ab aabbaa=abba abbabbaa=aabbabba