#383 ⟨
a
,
b
|
aabbaab
=
a
⟩
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Monoid enumeration
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#382
⟨
a
,
b
|
aababba
=
b
⟩
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#384
⟨
a
,
b
|
aabbaab
=
b
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
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
b
=
a
and
a
⋅ 1 =
a
, however
a
b
2
a
2
b
≠ 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
b
<
a
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
2
b
2
a
⇒
ab
a
2
b
[2]
2.
a
3
bab
⇒ (
aba
)
2
[4]
3.
(
aba
)
2
b
⇒
a
[3]
# ab:aabbaab=a ba - - aabba=abaab aaabab=abaaba abaabab=a