#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
⟩
Properties
Presentation has sum-of-sides 11
Infinite non-cancellative commutative monoid
Commutative Gröbner basis: ⟨
a
,
b
|
ab
=
a
,
a
6
=
a
⟩
Not cancellative, because multiplication by
a
is not injective:
b
≠ 1
a
⋅
b
=
a
a
⋅ 1 =
a
Cancellative quotient is isomorphic to ℤ
5
Grothendieck group is isomorphic to ℤ
5
Group of units is isomorphic to ℤ
1
3 Archimedian components:
1,
a
,
b
Complete 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
ab
⇒
a
ba
⇒
a
a
6
⇒
a
# ab:ab=a,baaaaaa=a ab ab=a ba=a aaaaaa=a
Staircase diagram
Right Cayley graph (truncated)
