#1110 ⟨
a
,
b
|
abaaab
=
bbb
⟩
Up:
Monoids with two generators and one relation
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#1109
⟨
a
,
b
|
abaaab
=
bba
⟩
Next:
#1111
⟨
a
,
b
|
abaaba
=
aaa
⟩
Quick links
Properties
Rewriting system
Properties
Presentation has sum-of-sides 9
Infinite non-cancellative non-commutative monoid
Not right cancellative, because right multiplication by
b
is not injective:
ab
a
3
⋅
b
=
b
3
and
b
2
⋅
b
=
b
3
, however
ab
a
3
≠
b
2
Enveloping group: ⟨
a
,
b
|
aaabab
-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
b
<
a
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ab
a
3
b
⇒
b
3
[1]
2.
ab
a
2
b
3
⇒
b
3
a
3
b
[2]
# ab:abaaab=bbb ba abaaab=bbb abaabbb=bbbaaab