#1143 ⟨
a
,
b
|
aaaaa
=
abab
⟩
Up:
Monoid enumeration
Prev:
#1142
⟨
a
,
b
|
aaaaa
=
aabb
⟩
Next:
#1144
⟨
a
,
b
|
aaaaa
=
abba
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 9
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
a
4
= (
ab
)
2
and
a
⋅
bab
= (
ab
)
2
, however
a
4
≠
bab
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
6
b
⇒
ab
a
5
[2]
2.
(
ab
)
2
⇒
a
5
[1]
# ab:aaaaa=abab a/b - - aaaaaab=abaaaaa abab=aaaaa