#1489 ⟨
a
,
b
|
abaaabbbab
=1⟩
Up:
Monoids with two generators and one relation
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#1481
⟨
a
,
b
|
abaaaabbab
=1⟩
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#1496
⟨
a
,
b
|
abaababbab
=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
3
b
3
aba
=
a
and
a
⋅ 1 =
a
, however
b
a
3
b
3
aba
≠ 1
Not right cancellative, because right multiplication by
b
a
3
b
3
ab
is not injective:
b
a
3
b
3
aba
⋅
b
a
3
b
3
ab
=
b
a
3
b
3
ab
and 1 ⋅
b
a
3
b
3
ab
=
b
a
3
b
3
ab
, however
b
a
3
b
3
aba
≠ 1
Enveloping group: ⟨
a
,
b
|
aaababab
-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
3
b
3
⇒
a
3
b
3
ab
[2]
2.
a
3
b
3
(
ab
)
2
⇒ 1
[3]
# ab:abaaabbbab=1 ab abaaabbb=aaabbbab aaabbbabab=1