#1990 ⟨
a
,
b
|
aabbaabb
=
ab
⟩
Up:
Monoids with two generators and one relation
Prev:
#1989
⟨
a
,
b
|
aabbaabb
=
aa
⟩
Next:
#1991
⟨
a
,
b
|
aabbaabb
=
ba
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 10
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
a
b
2
a
2
b
2
=
ab
and
a
⋅
b
=
ab
, however
a
b
2
a
2
b
2
≠
b
Not right cancellative, because right multiplication by
b
is not injective:
a
2
b
2
a
2
b
⋅
b
=
ab
and
a
⋅
b
=
ab
, however
a
2
b
2
a
2
b
≠
a
Enveloping group: ⟨
a
,
b
|
aabab
-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 shortlex with
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ab
a
2
b
2
⇒
a
2
b
2
ab
[2]
2.
(
a
2
b
2
)
2
⇒
ab
[1]
# ab:aabbaabb=ab ab abaabb=aabbab aabbaabb=ab