#1809 ⟨
a
,
b
|
abbababba
=
a
⟩
Up:
Monoids with two generators and one relation
Prev:
#1808
⟨
a
,
b
|
abbabaaab
=
b
⟩
Next:
#1810
⟨
a
,
b
|
abbababba
=
b
⟩
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
⋅
b
(
ba
)
2
b
2
a
=
a
and
a
⋅ 1 =
a
, however
b
(
ba
)
2
b
2
a
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
ab
(
ba
)
2
b
2
⋅
a
=
a
and 1 ⋅
a
=
a
, however
ab
(
ba
)
2
b
2
≠ 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.
ab
(
ba
)
2
⇒
aba
b
2
a
[2]
2.
aba
(
b
2
a
)
2
⇒
a
[3]
# ab:abbababba=a ab abbaba=ababba ababbabba=a