#1170 ⟨
a
,
b
|
aaaba
=
abab
⟩
Up:
Monoids with two generators and one relation
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#1169
⟨
a
,
b
|
aaaba
=
abaa
⟩
Next:
#1171
⟨
a
,
b
|
aaaba
=
abba
⟩
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
ba
= (
ab
)
2
and
a
⋅
bab
= (
ab
)
2
, however
a
2
ba
≠
bab
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 recursive path with deg(
a
) = 0; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
(
ab
)
2
⇒
a
3
ba
[1]
2.
a
(
a
2
b
)
2
⇒
ab
a
3
ba
[2]
# ab:aaaba=abab a/b abab=aaaba aaabaab=abaaaba