#70 ⟨
a
,
b
|
ababab
=1⟩
Up:
Monoids with two generators and one relation
Prev:
#69
⟨
a
,
b
|
abaaba
=1⟩
Next:
#71
⟨
a
,
b
|
ababba
=1⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 6
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅ (
ba
)
3
=
a
and
a
⋅ 1 =
a
, however (
ba
)
3
≠ 1
Not right cancellative, because right multiplication by
b
(
ab
)
2
is not injective:
(
ba
)
3
⋅
b
(
ab
)
2
=
b
(
ab
)
2
and 1 ⋅
b
(
ab
)
2
=
b
(
ab
)
2
, however (
ba
)
3
≠ 1
Enveloping group is isomorphic to ℤ
3
∗ 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.
(
ab
)
3
⇒ 1
[1]
# ab:ababab=1 ab ababab=1