#2678 ⟨
a
,
b
|
ababa
=
aaaaa
⟩
Up:
Monoids with two generators and one relation
Prev:
#2677
⟨
a
,
b
|
abaab
=
abaaa
⟩
Next:
#2679
⟨
a
,
b
|
ababa
=
aaaab
⟩
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
⋅ (
ba
)
2
=
a
5
and
a
⋅
a
4
=
a
5
, however (
ba
)
2
≠
a
4
Not right cancellative, because right multiplication by
a
is not injective:
(
ab
)
2
⋅
a
=
a
5
and
a
4
⋅
a
=
a
5
, however (
ab
)
2
≠
a
4
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
(
ba
)
2
⇒
a
5
[1]
2.
ab
a
5
⇒
a
5
ba
[2]
# ab:ababa=aaaaa ab ababa=aaaaa abaaaaa=aaaaaba