#568 ⟨
a
,
b
|
abab
=
aaba
⟩
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Monoids with two generators and one relation
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#567
⟨
a
,
b
|
abab
=
aaab
⟩
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#569
⟨
a
,
b
|
abab
=
aabb
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 8
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
bab
=
a
2
ba
and
a
⋅
aba
=
a
2
ba
, however
bab
≠
aba
Enveloping group: ⟨
a
,
b
|
aaabb
⟩
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
Auxiliary generators:
c
=
ab
Reduction order:
Left-to-right shortlex with
a
<
b
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
Morphocompletion
: how the auxiliary generators were found.
#
Rule
Proof
1.
ab
⇒
c
[2]
2.
aca
⇒
c
2
[4]
3.
c
2
b
⇒
a
c
2
[5]
4.
c
3
a
⇒
a
c
3
[6]
# ab:abab=aaba abc ab=c morph:2/0 ab=c aca=cc ccb=acc ccca=accc