#2238 ⟨
a
,
b
|
aababab
=
aab
⟩
Up:
Monoids with two generators and one relation
Prev:
#2237
⟨
a
,
b
|
aababab
=
aaa
⟩
Next:
#2239
⟨
a
,
b
|
aababab
=
aba
⟩
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
)
3
=
a
2
b
and
a
⋅
ab
=
a
2
b
, however (
ab
)
3
≠
ab
Not right cancellative, because right multiplication by
b
is not injective:
a
2
(
ba
)
2
⋅
b
=
a
2
b
and
a
2
⋅
b
=
a
2
b
, however
a
2
(
ba
)
2
≠
a
2
Enveloping group is isomorphic to ℤ
2
∗ 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
(
ab
)
3
⇒
a
2
b
[1]
# ab:aababab=aab ab aababab=aab