#2682 ⟨
a
,
b
|
ababa
=
aabaa
⟩
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Monoid enumeration
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#2681
⟨
a
,
b
|
ababa
=
aaabb
⟩
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#2683
⟨
a
,
b
|
ababa
=
aabab
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅ (
ba
)
2
=
a
2
b
a
2
and
a
⋅
ab
a
2
=
a
2
b
a
2
, however (
ba
)
2
≠
ab
a
2
Not right cancellative, because right multiplication by
a
is not injective:
(
ab
)
2
⋅
a
=
a
2
b
a
2
and
a
2
ba
⋅
a
=
a
2
b
a
2
, however (
ab
)
2
≠
a
2
ba
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
b
<
a
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
2
b
a
2
⇒
a
(
ba
)
2
[1]
2.
a
2
(
ba
)
3
⇒ (
ab
)
3
a
2
[2]
# ab:ababa=aabaa ba - - aabaa=ababa aabababa=abababaa