#1035 ⟨
a
,
b
|
aaabaa
=
aba
⟩
Up:
Monoids with two generators and one relation
Prev:
#1034
⟨
a
,
b
|
aaabaa
=
aab
⟩
Next:
#1036
⟨
a
,
b
|
aaabaa
=
abb
⟩
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
⋅
a
2
b
a
2
=
aba
and
a
⋅
ba
=
aba
, however
a
2
b
a
2
≠
ba
Not right cancellative, because right multiplication by
a
is not injective:
a
3
ba
⋅
a
=
aba
and
ab
⋅
a
=
aba
, however
a
3
ba
≠
ab
Enveloping group: ⟨
a
,
b
|
aabab
-1
⟩
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
b
<
a
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
3
b
a
2
⇒
aba
[1]
2.
a
3
(
ba
)
2
⇒
a
(
b
a
2
)
2
[2]
# ab:aaabaa=aba ba aaabaa=aba aaababa=abaabaa