#12413 ⟨
a
,
b
|
aaba
=
bb
,
abbb
=
b
⟩
Up:
Monoid enumeration
Prev:
#12403
⟨
a
,
b
|
aaba
=
ba
,
bbbb
=
b
⟩
Next:
#12415
⟨
a
,
b
|
aaba
=
bb
,
baaa
=
b
⟩
Contents
Properties
Commutative structure
Staircase diagram
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative commutative monoid
Element
a
has infinite order
Not cancellative, because multiplication by
b
is not injective:
b
⋅
a
7
=
b
and
b
⋅ 1 =
b
, however
a
7
≠ 1
Commutative structure
Gröbner basis: ⟨
a
,
b
|
b
8
=
b
,
ab
=
b
6
⟩
Cancellative quotient is isomorphic to ℤ
7
Enveloping group is isomorphic to ℤ
7
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 recursive path with deg(
a
) = 0; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
b
a
7
⇒
b
[11]
2.
ab
⇒
ba
[10]
3.
b
2
⇒
b
a
3
[12]
# ab:aaba=bb,abbb=b a/b - - baaaaaaa=b ab=ba bb=baaa
