#1353 ⟨
a
,
b
|
aaababaabb
=1⟩
Up:
Monoids with two generators and one relation
Prev:
#1350
⟨
a
,
b
|
aaabaabbba
=1⟩
Next:
#1355
⟨
a
,
b
|
aaabababab
=1⟩
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
(
ab
)
2
a
2
b
2
a
=
a
and
a
⋅ 1 =
a
, however
a
(
ab
)
2
a
2
b
2
a
≠ 1
Not right cancellative, because right multiplication by
a
(
ab
)
2
a
2
b
2
is not injective:
a
(
ab
)
2
a
2
b
2
a
⋅
a
(
ab
)
2
a
2
b
2
=
a
(
ab
)
2
a
2
b
2
and 1 ⋅
a
(
ab
)
2
a
2
b
2
=
a
(
ab
)
2
a
2
b
2
, however
a
(
ab
)
2
a
2
b
2
a
≠ 1
Enveloping group: ⟨
a
,
b
|
aabab
-1
abb
⟩
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
3
bab
a
2
b
2
⇒ 1
[1]
# ab:aaababaabb=1 ab aaababaabb=1