#2185 ⟨
a
,
b
|
aaabbaa
=
aba
⟩
Up:
Monoids with two generators and one relation
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#2184
⟨
a
,
b
|
aaabbaa
=
aab
⟩
Next:
#2186
⟨
a
,
b
|
aaabbaa
=
abb
⟩
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
⋅
a
2
b
2
a
2
=
aba
and
a
⋅
ba
=
aba
, however
a
2
b
2
a
2
≠
ba
Not right cancellative, because right multiplication by
a
is not injective:
a
3
b
2
a
⋅
a
=
aba
and
ab
⋅
a
=
aba
, however
a
3
b
2
a
≠
ab
Enveloping group: ⟨
a
,
b
|
aabab
-1
b
-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
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
3
b
2
a
2
⇒
aba
[1]
2.
ab
a
2
b
2
a
2
⇒
a
3
b
(
ba
)
2
[2]
# ab:aaabbaa=aba ab aaabbaa=aba abaabbaa=aaabbaba