#630 ⟨
a
,
b
|
aaabaabab
=1⟩
Up:
Monoids with two generators and one relation
Prev:
#628
⟨
a
,
b
|
aaabaaabb
=1⟩
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#631
⟨
a
,
b
|
aaabaabba
=1⟩
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
⋅ (
a
2
b
)
2
aba
=
a
and
a
⋅ 1 =
a
, however (
a
2
b
)
2
aba
≠ 1
Not right cancellative, because right multiplication by (
a
2
b
)
2
ab
is not injective:
(
a
2
b
)
2
aba
⋅ (
a
2
b
)
2
ab
= (
a
2
b
)
2
ab
and 1 ⋅ (
a
2
b
)
2
ab
= (
a
2
b
)
2
ab
, however (
a
2
b
)
2
aba
≠ 1
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.
a
(
a
2
b
)
2
ab
⇒ 1
[1]
# ab:aaabaabab=1 ab aaabaabab=1