#6347 ⟨
a
,
b
|
aab
=
a
,
abbba
=
b
⟩
Up:
Monoids with two generators and two relations
Prev:
#6345
⟨
a
,
b
|
aab
=
a
,
abbab
=
b
⟩
Next:
#6361
⟨
a
,
b
|
aab
=
a
,
babab
=
b
⟩
Properties
Presentation has sum-of-sides 10
Infinite non-cancellative commutative monoid
Commutative Gröbner basis: ⟨
a
,
b
|
a
b
2
=
b
,
a
2
b
=
a
⟩
Not cancellative, because multiplication by
ab
is not injective:
ab
≠ 1
ab
⋅
ab
=
ab
ab
⋅ 1 =
ab
Grothendieck group is isomorphic to ℤ
Group of units is isomorphic to ℤ
1
2 Archimedian components:
1,
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
ba
⇒
ab
a
2
b
⇒
a
a
b
2
⇒
b
# ab:aab=a,abbba=b ab ba=ab aab=a abb=b
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
6471
⟨
a
,
b
|
aba
=
a
,
abbba
=
b
⟩
φ
(
a
) =
a
,
φ
(
b
) =
b
