#5546 ⟨
a
,
b
|
aaabaa
=
abbab
⟩
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Monoids with two generators and one relation
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#5545
⟨
a
,
b
|
aaabaa
=
abbaa
⟩
Next:
#5547
⟨
a
,
b
|
aaabaa
=
abbba
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 11
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
a
2
b
a
2
=
a
b
2
ab
and
a
⋅
b
2
ab
=
a
b
2
ab
, however
a
2
b
a
2
≠
b
2
ab
Enveloping group: ⟨
a
,
b
|
aabaab
-1
ab
-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 recursive path with deg(
a
) = 0; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
b
2
ab
⇒
a
3
b
a
2
[1]
2.
a
(
a
2
b
)
2
ab
⇒
a
b
2
a
3
b
a
2
[2]
# ab:aaabaa=abbab a/b abbab=aaabaa aaabaabab=abbaaabaa