#20104 ⟨
a
,
b
|
aab
=
b
,
abbb
=
aaa
⟩
Up:
Monoid enumeration
Prev:
#20103
⟨
a
,
b
|
aab
=
b
,
abba
=
bbb
⟩
Next:
#20111
⟨
a
,
b
|
aab
=
b
,
abbb
=
bbb
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
ab
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅
a
2
=
b
and
b
⋅ 1 =
b
, however
a
2
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
a
2
⋅
b
=
b
and 1 ⋅
b
=
b
, however
a
2
≠ 1
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.
a
5
⇒
a
3
[6]
2.
b
a
2
⇒
b
[8]
3.
a
2
b
⇒
b
[1]
4.
b
3
⇒
a
4
[3]
# ab:aab=b,abbb=aaa a/b - - aaaaa=aaa baa=b aab=b bbb=aaaa