#1481 ⟨
a
,
b
|
abaaaabbab
=1⟩
Up:
Monoids with two generators and one relation
Prev:
#1473
⟨
a
,
b
|
aabbbbbbab
=1⟩
Next:
#1489
⟨
a
,
b
|
abaaabbbab
=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
⋅
b
a
4
b
(
ba
)
2
=
a
and
a
⋅ 1 =
a
, however
b
a
4
b
(
ba
)
2
≠ 1
Not right cancellative, because right multiplication by
b
a
4
b
2
ab
is not injective:
b
a
4
b
(
ba
)
2
⋅
b
a
4
b
2
ab
=
b
a
4
b
2
ab
and 1 ⋅
b
a
4
b
2
ab
=
b
a
4
b
2
ab
, however
b
a
4
b
(
ba
)
2
≠ 1
Enveloping group: ⟨
a
,
b
|
aaabab
-1
b
-1
b
-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
4
b
2
⇒
a
4
b
2
ab
[2]
2.
a
4
b
2
(
ab
)
2
⇒ 1
[3]
# ab:abaaaabbab=1 ab abaaaabb=aaaabbab aaaabbabab=1