#4617 ⟨
a
,
b
|
aaaa
=
a
,
aaba
=
b
⟩
Up:
Monoids with two generators and two relations
Prev:
#4616
⟨
a
,
b
|
aaaa
=
a
,
aaba
=
a
⟩
Next:
#4618
⟨
a
,
b
|
aaaa
=
a
,
aabb
=
a
⟩
Quick links
Properties
Staircase diagram
Rewriting system
Properties
Presentation has sum-of-sides 10
Infinite non-cancellative commutative monoid
Not cancellative, because multiplication by
ab
is not injective:
ab
⋅
a
3
=
ab
and
ab
⋅ 1 =
ab
, however
a
3
≠ 1
Commutative Gröbner basis: ⟨
a
,
b
|
a
4
=
a
,
a
3
b
=
b
⟩
Cancellative quotient is isomorphic to ℤ
3
⊕ ℕ
Enveloping group is isomorphic to ℤ
3
⊕ ℤ
Group of units is isomorphic to ℤ
1
3 Archimedian components:
1,
a
,
b
Staircase diagram
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
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ba
⇒
ab
[4]
2.
a
4
⇒
a
[1]
3.
a
3
b
⇒
b
[3]
# ab:aaaa=a,aaba=b ab ba=ab aaaa=a aaab=b
