#7119 ⟨
a
,
b
|
ba
=
ab
,
aaaa
=
ab
⟩
Up:
Monoids with two generators and two relations
Prev:
#7118
⟨
a
,
b
|
ba
=
ab
,
aaaa
=
aa
⟩
Next:
#7120
⟨
a
,
b
|
ba
=
ab
,
aaaa
=
bb
⟩
Properties
Presentation has sum-of-sides 10
Infinite non-cancellative commutative monoid
Commutative Gröbner basis: ⟨
a
,
b
|
ab
=
a
4
⟩
Not cancellative, because multiplication by
a
4
is not injective:
a
3
≠
b
a
4
⋅
a
3
=
a
7
a
4
⋅
b
=
a
7
Cancellative quotient is isomorphic to ℕ
Grothendieck group is isomorphic to ℤ
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
ba
⇒
a
4
ab
⇒
a
4
# ab:ba=ab,aaaa=ab a/b ba=aaaa ab=aaaa
Staircase diagram
Right Cayley graph (truncated)
