#2484 ⟨
a
,
b
|
aabaab
=
abab
⟩
Up:
Monoids with two generators and one relation
Prev:
#2483
⟨
a
,
b
|
aabaab
=
abaa
⟩
Next:
#2485
⟨
a
,
b
|
aabaab
=
abba
⟩
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
⋅
ab
a
2
b
= (
ab
)
2
and
a
⋅
bab
= (
ab
)
2
, however
ab
a
2
b
≠
bab
Not right cancellative, because right multiplication by
b
is not injective:
a
2
b
a
2
⋅
b
= (
ab
)
2
and
aba
⋅
b
= (
ab
)
2
, however
a
2
b
a
2
≠
aba
Enveloping group: ⟨
a
,
b
|
aabb
⟩
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
2
b
)
2
⇒ (
ab
)
2
[1]
2.
a
(
ab
)
3
⇒ (
ab
)
2
a
2
b
[2]
# ab:aabaab=abab ba aabaab=abab aababab=ababaab