#1104 ⟨
a
,
b
|
abaaab
=
aab
⟩
Up:
Monoids with two generators and one relation
Prev:
#1103
⟨
a
,
b
|
abaaab
=
aaa
⟩
Next:
#1105
⟨
a
,
b
|
abaaab
=
aba
⟩
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
⋅
b
a
3
b
=
a
2
b
and
a
⋅
ab
=
a
2
b
, however
b
a
3
b
≠
ab
Not right cancellative, because right multiplication by
b
is not injective:
ab
a
3
⋅
b
=
a
2
b
and
a
2
⋅
b
=
a
2
b
, however
ab
a
3
≠
a
2
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
Auxiliary generators:
c
=
aaab
Reduction order:
Left-to-right shortlex with
b
<
c
<
a
Certificate
: derivations of all rewriting rules from the defining relations.
Morphocompletion
: how the auxiliary generators were found.
#
Rule
Proof
1.
ac
⇒
c
2
[5]
2.
a
2
b
⇒
abc
[3]
3.
ab
c
2
⇒
c
[4]
# ab:abaaab=aab bca aaab=c morph:4/1 ac=cc aab=abc abcc=c