#285 ⟨
a
,
b
|
aaabaaab
=1⟩
Up:
Monoids with two generators and one relation
Prev:
#284
⟨
a
,
b
|
aaaabbbb
=1⟩
Next:
#287
⟨
a
,
b
|
aaabaabb
=1⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 8
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅ (
a
2
ba
)
2
=
a
and
a
⋅ 1 =
a
, however (
a
2
ba
)
2
≠ 1
Not right cancellative, because right multiplication by
a
2
b
a
3
b
is not injective:
(
a
2
ba
)
2
⋅
a
2
b
a
3
b
=
a
2
b
a
3
b
and 1 ⋅
a
2
b
a
3
b
=
a
2
b
a
3
b
, however (
a
2
ba
)
2
≠ 1
Enveloping group is isomorphic to ℤ
2
∗ ℤ
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
b
)
2
⇒ 1
[1]
# ab:aaabaaab=1 ab aaabaaab=1