#926 ⟨
a
,
b
|
aabaaab
=
ab
⟩
Up:
Monoids with two generators and one relation
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#925
⟨
a
,
b
|
aabaaab
=
aa
⟩
Next:
#927
⟨
a
,
b
|
aabaaab
=
ba
⟩
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
⋅
ab
a
3
b
=
ab
and
a
⋅
b
=
ab
, however
ab
a
3
b
≠
b
Not right cancellative, because right multiplication by
b
is not injective:
a
2
b
a
3
⋅
b
=
ab
and
a
⋅
b
=
ab
, however
a
2
b
a
3
≠
a
Enveloping group is isomorphic to 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
a
2
b
⇒
a
(
ab
)
2
[3]
2.
ab
a
3
b
⇒
a
3
bab
[5]
3.
a
4
bab
⇒
ab
[4]
# ab:aabaaab=ab ab abaab=aabab abaaab=aaabab aaaabab=ab