#24460 ⟨
a
,
b
|
ab
=
a
,
baaaaaa
=
a
⟩
Up:
Monoids with two generators and two relations
Prev:
#24332
⟨
a
,
b
|
ab
=
a
,
aaaaaaa
=
a
⟩
Next:
#24524
⟨
a
,
b
|
ab
=
a
,
bbaaaaa
=
a
⟩
Quick links
Properties
Staircase diagram
Rewriting system
Properties
Presentation has sum-of-sides 11
Infinite non-cancellative commutative monoid
Not cancellative, because multiplication by
a
is not injective:
a
⋅
b
=
a
and
a
⋅ 1 =
a
, however
b
≠ 1
Commutative Gröbner basis: ⟨
a
,
b
|
ab
=
a
,
a
6
=
a
⟩
Cancellative quotient is isomorphic to ℤ
5
Enveloping group is isomorphic to ℤ
5
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.
ab
⇒
a
[1]
2.
ba
⇒
a
[6]
3.
a
6
⇒
a
[5]
# ab:ab=a,baaaaaa=a ab ab=a ba=a aaaaaa=a
