#5555 ⟨
a
,
b
|
aaabaa
=
babba
⟩
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Monoids with two generators and one relation
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#5554
⟨
a
,
b
|
aaabaa
=
babab
⟩
Next:
#5556
⟨
a
,
b
|
aaabaa
=
babbb
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 11
Infinite non-cancellative non-commutative monoid
Not right cancellative, because right multiplication by
a
is not injective:
a
3
ba
⋅
a
=
ba
b
2
a
and
ba
b
2
⋅
a
=
ba
b
2
a
, however
a
3
ba
≠
ba
b
2
Enveloping group: ⟨
a
,
b
|
aaabab
-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:
Right-to-left recursive path with deg(
a
) = 0; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ba
b
2
a
⇒
a
3
b
a
2
[1]
2.
b
(
ab
a
2
)
2
⇒
a
3
b
a
2
b
2
a
[2]
# ab:aaabaa=babba reversed:a/b babba=aaabaa babaaabaa=aaabaabba