#658 ⟨
a
,
b
|
ab
=
aa
,
ba
=
aa
⟩
Up:
Monoids with two generators and two relations
Prev:
#657
⟨
a
,
b
|
bb
=
aa
,
aba
=
b
⟩
Next:
#660
⟨
a
,
b
|
ab
=
aa
,
bb
=
aa
⟩
Properties
Presentation has sum-of-sides 8
Infinite non-cancellative commutative monoid
Commutative Gröbner basis: ⟨
a
,
b
|
ab
=
a
2
⟩
Not cancellative, because multiplication by
a
2
is not injective:
a
≠
b
a
2
⋅
a
=
a
3
a
2
⋅
b
=
a
3
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 shortlex with
a
<
b
ab
⇒
a
2
ba
⇒
a
2
# ab:ab=aa,ba=aa ab ab=aa ba=aa
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
8
659
⟨
a
,
b
|
ab
=
aa
,
ba
=
ab
⟩
φ
(
a
) =
a
,
φ
(
b
) =
b
