#4617 ⟨
a
,
b
|
aaaa
=
a
,
aaba
=
b
⟩
Up:
Monoid enumeration
Prev:
#4616
⟨
a
,
b
|
aaaa
=
a
,
aaba
=
a
⟩
Next:
#4618
⟨
a
,
b
|
aaaa
=
a
,
aabb
=
a
⟩
Contents
Properties
Commutative structure
Staircase diagram
Rewriting system
Properties
Sum of relation sides is 10
Infinite non-cancellative commutative monoid
Element
b
has infinite order
Not cancellative, because multiplication by
a
is not injective:
a
⋅
a
3
=
a
and
a
⋅ 1 =
a
, however
a
3
≠ 1
Commutative structure
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:
Enter a word above to compute its normal form. Tips:
Lowercase letters stand for generators.
Spaces are ignored.
Numbers repeat the previous letter, e.g.
b90
.
Strategy:
Leftmost
Rightmost
Result:
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
