#1260 ⟨
a
,
b
|
ababa
=
abba
⟩
Up:
Monoids with two generators and one relation
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#1259
⟨
a
,
b
|
ababa
=
abab
⟩
Next:
#1261
⟨
a
,
b
|
ababa
=
abbb
⟩
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
⋅ (
ba
)
2
=
a
b
2
a
and
a
⋅
b
2
a
=
a
b
2
a
, however (
ba
)
2
≠
b
2
a
Not right cancellative, because right multiplication by
a
is not injective:
(
ab
)
2
⋅
a
=
a
b
2
a
and
a
b
2
⋅
a
=
a
b
2
a
, however (
ab
)
2
≠
a
b
2
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.
a
(
ba
)
2
⇒
a
b
2
a
[1]
2.
ab
(
ba
)
2
⇒
aba
b
2
a
[2]
# ab:ababa=abba ab ababa=abba abbaba=ababba