#15944 ⟨
a
,
b
|
aba
=
bb
,
bbbbb
=
b
⟩
Up:
Monoids with two generators and two relations
Prev:
#15941
⟨
a
,
b
|
aba
=
bb
,
bbabb
=
a
⟩
Next:
#15945
⟨
a
,
b
|
aaa
=
aa
,
aaaa
=
aa
⟩
Quick links
Properties
Staircase diagram
Rewriting system
Properties
Presentation has sum-of-sides 11
Infinite non-cancellative commutative monoid
Not cancellative, because multiplication by
ab
is not injective:
ab
⋅
a
2
=
a
b
2
and
ab
⋅
b
=
a
b
2
, however
a
2
≠
b
Commutative Gröbner basis: ⟨
a
,
b
|
a
2
b
=
b
2
,
b
5
=
b
⟩
Cancellative quotient is isomorphic to ℤ
8
Enveloping group is isomorphic to ℤ
8
Group of units is isomorphic to ℤ
1
3 Archimedian components:
1,
a
,
b
Staircase diagram
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.
ba
⇒
ab
[5]
2.
a
2
b
⇒
b
2
[7]
3.
b
5
⇒
b
[2]
# ab:aba=bb,bbbbb=b ab ba=ab aab=bb bbbbb=b
