#7118 ⟨
a
,
b
|
ba
=
ab
,
aaaa
=
aa
⟩
Up:
Monoids with two generators and two relations
Prev:
#7117
⟨
a
,
b
|
ab
=
aa
,
bbbb
=
bb
⟩
Next:
#7119
⟨
a
,
b
|
ba
=
ab
,
aaaa
=
ab
⟩
Properties
Presentation has sum-of-sides 10
Isomorphic to ℕ
(4 = 2)
⊕ ℕ
Infinite non-cancellative commutative monoid
Commutative Gröbner basis: ⟨
a
,
b
|
a
4
=
a
2
⟩
Not cancellative, because multiplication by
a
2
b
2
is not injective:
a
2
≠ 1
a
2
b
2
⋅
a
2
=
a
2
b
2
a
2
b
2
⋅ 1 =
a
2
b
2
Cancellative quotient is isomorphic to ℤ
2
⊕ ℕ
Grothendieck group is isomorphic to ℤ
2
⊕ ℤ
Group of units is isomorphic to ℤ
1
4 Archimedian components:
1,
a
,
b
,
ab
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
ba
⇒
ab
a
4
⇒
a
2
# ab:ba=ab,aaaa=aa ab ba=ab aaaa=aa
Staircase diagram
Right Cayley graph (truncated)
