#41 ⟨
a
,
b
|
abab
=
a
⟩
Up:
Monoids with two generators and one relation
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#40
⟨
a
,
b
|
aabb
=
a
⟩
Next:
#42
⟨
a
,
b
|
abba
=
a
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 5
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
bab
=
a
and
a
⋅ 1 =
a
, however
bab
≠ 1
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
Reduction order:
Left-to-right shortlex with
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
aba
⇒
a
2
b
[2]
2.
a
2
b
2
⇒
a
[3]
# ab:abab=a ab aba=aab aabb=a