#303 ⟨
a
,
b
|
aabaabab
=1⟩
Up:
Monoids with two generators and one relation
Prev:
#301
⟨
a
,
b
|
aabaaabb
=1⟩
Next:
#307
⟨
a
,
b
|
aabababb
=1⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 8
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅ (
aba
)
2
ba
=
a
and
a
⋅ 1 =
a
, however (
aba
)
2
ba
≠ 1
Not right cancellative, because right multiplication by (
aba
)
2
b
is not injective:
(
aba
)
2
ba
⋅ (
aba
)
2
b
= (
aba
)
2
b
and 1 ⋅ (
aba
)
2
b
= (
aba
)
2
b
, however (
aba
)
2
ba
≠ 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
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
(
a
2
b
)
2
ab
⇒ 1
[1]
# ab:aabaabab=1 ab aabaabab=1