#12500 ⟨
a
,
b
|
abab
=
aa
,
abbb
=
a
⟩
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Monoid enumeration
Prev:
#12499
⟨
a
,
b
|
abab
=
aa
,
abba
=
b
⟩
Next:
#12508
⟨
a
,
b
|
abab
=
aa
,
babb
=
a
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
b
3
=
a
and
a
⋅ 1 =
a
, however
b
3
≠ 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(
b
) = 0; deg(
a
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
b
3
⇒
a
[2]
2.
aba
⇒
a
2
b
2
[4]
3.
a
2
b
2
a
⇒
a
3
b
[5]
# ab:abab=aa,abbb=a b/a - - abbb=a aba=aabb aabba=aaab