#1083 ⟨
a
,
b
|
aabbaa
=
aba
⟩
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Monoid enumeration
Prev:
#1082
⟨
a
,
b
|
aabbaa
=
aab
⟩
Next:
#1084
⟨
a
,
b
|
aabbaa
=
abb
⟩
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
b
2
a
2
=
aba
and
a
⋅
ba
=
aba
, however
a
b
2
a
2
≠
ba
Not right cancellative, because right multiplication by
a
is not injective:
a
2
b
2
a
⋅
a
=
aba
and
ab
⋅
a
=
aba
, however
a
2
b
2
a
≠
ab
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
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
2
b
2
a
2
⇒
aba
[1]
2.
aba
b
2
a
2
⇒
a
2
b
(
ba
)
2
[2]
# ab:aabbaa=aba ab - - aabbaa=aba ababbaa=aabbaba