#248 ⟨
a
,
b
|
aaba
=
aab
⟩
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Monoids with two generators and one relation
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#247
⟨
a
,
b
|
aaba
=
aaa
⟩
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#249
⟨
a
,
b
|
aaba
=
aba
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 7
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
aba
=
a
2
b
and
a
⋅
ab
=
a
2
b
, however
aba
≠
ab
Enveloping group is isomorphic to ℤ
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
=
aab
Reduction order:
Left-to-right shortlex with
a
<
c
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ca
⇒
c
[4]
2.
cb
⇒
c
2
[5]
3.
a
2
b
⇒
c
[2]
# ab:aaba=aab acb aab=c magic:0 ca=c cb=cc aab=c