#12186 ⟨
a
,
b
|
aaaa
=
ab
,
babb
=
a
⟩
Up:
Monoids with two generators and two relations
Prev:
#12183
⟨
a
,
b
|
aaaa
=
ab
,
baab
=
b
⟩
Next:
#12187
⟨
a
,
b
|
aaaa
=
ab
,
babb
=
b
⟩
Properties
Presentation has sum-of-sides 11
Infinite non-cancellative commutative monoid
Commutative Gröbner basis: ⟨
a
,
b
|
a
4
=
ab
,
a
b
3
=
a
⟩
Not cancellative, because multiplication by
ab
is not injective:
b
3
≠ 1
ab
⋅
b
3
=
ab
ab
⋅ 1 =
ab
Cancellative quotient is isomorphic to ℤ
9
Grothendieck group is isomorphic to ℤ
9
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 recursive path with deg(
a
) = 0; deg(
b
) = 1
a
10
⇒
a
ba
⇒
a
4
ab
⇒
a
4
# ab:aaaa=ab,babb=a a/b aaaaaaaaaa=a ba=aaaa ab=aaaa
Staircase diagram
Right Cayley graph (truncated)
