#929 ⟨
a
,
b
|
aabaaba
=
aa
⟩
Up:
Monoids with two generators and one relation
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#928
⟨
a
,
b
|
aabaaab
=
bb
⟩
Next:
#930
⟨
a
,
b
|
aabaaba
=
ab
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 9
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅ (
aba
)
2
=
a
2
and
a
⋅
a
=
a
2
, however (
aba
)
2
≠
a
Not right cancellative, because right multiplication by
a
is not injective:
(
a
2
b
)
2
⋅
a
=
a
2
and
a
⋅
a
=
a
2
, however (
a
2
b
)
2
≠
a
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
a
2
⇒
a
3
ba
[2]
2.
a
3
(
ba
)
2
⇒
a
2
[3]
# ab:aabaaba=aa ab aabaa=aaaba aaababa=aa