#646 ⟨
a
,
b
|
ba
=
ab
,
aaa
=
a
⟩
Up:
Monoids with two generators and two relations
Prev:
#645
⟨
a
,
b
|
ab
=
aa
,
bbb
=
b
⟩
Next:
#652
⟨
a
,
b
|
bb
=
aa
,
aaa
=
a
⟩
Properties
Presentation has sum-of-sides 8
Isomorphic to ℕ
(3 = 1)
⊕ ℕ
Infinite non-cancellative commutative monoid
Commutative Gröbner basis: ⟨
a
,
b
|
a
3
=
a
⟩
Not cancellative, because multiplication by
ab
is not injective:
a
2
≠ 1
ab
⋅
a
2
=
ab
ab
⋅ 1 =
ab
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
3
⇒
a
# ab:ba=ab,aaa=a ab ba=ab aaa=a
Staircase diagram
Right Cayley graph (truncated)
Other isomorphic instances
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
1 total
Σ
#
Presentation
Mapping
10
6531
⟨
a
,
b
|
aaa
=
a
,
aaab
=
ba
⟩
φ
(
a
) =
a
,
φ
(
b
) =
b
