#3449 ⟨
a
,
b
|
aaabaabbaa
=
a
⟩
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Monoid enumeration
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#3448
⟨
a
,
b
|
aaabaababb
=
b
⟩
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#3450
⟨
a
,
b
|
aaabaabbaa
=
b
⟩
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
⋅ (
a
2
b
)
2
b
a
2
=
a
and
a
⋅ 1 =
a
, however (
a
2
b
)
2
b
a
2
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
a
3
b
a
2
b
2
a
⋅
a
=
a
and 1 ⋅
a
=
a
, however
a
3
b
a
2
b
2
a
≠ 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 shortlex with
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ab
a
2
b
2
a
2
⇒ (
a
2
b
)
2
ba
[3]
2.
a
4
b
a
2
b
2
a
⇒
a
[4]
# ab:aaabaabbaa=a ab - - abaabbaa=aabaabba aaaabaabba=a