#5869 ⟨
a
,
b
|
ababab
=
abaab
⟩
Up:
Monoids with two generators and one relation
Prev:
#5868
⟨
a
,
b
|
ababab
=
abaaa
⟩
Next:
#5870
⟨
a
,
b
|
ababab
=
ababa
⟩
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
⋅
b
(
ab
)
2
=
ab
a
2
b
and
a
⋅
b
a
2
b
=
ab
a
2
b
, however
b
(
ab
)
2
≠
b
a
2
b
Not right cancellative, because right multiplication by
b
is not injective:
a
(
ba
)
2
⋅
b
=
ab
a
2
b
and
ab
a
2
⋅
b
=
ab
a
2
b
, however
a
(
ba
)
2
≠
ab
a
2
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
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
(
ab
)
3
⇒
ab
a
2
b
[1]
2.
(
ab
)
2
a
2
b
⇒ (
aba
)
2
b
[2]
# ab:ababab=abaab ab ababab=abaab ababaab=abaabab