#430 ⟨
a
,
b
|
aaabaa
=
ba
⟩
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Monoid enumeration
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#429
⟨
a
,
b
|
aaabaa
=
ab
⟩
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#431
⟨
a
,
b
|
aaabaa
=
bb
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not right cancellative, because right multiplication by
a
is not injective:
a
3
ba
⋅
a
=
ba
and
b
⋅
a
=
ba
, however
a
3
ba
≠
b
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
Auxiliary generators:
c
=
ba
Reduction order:
Left-to-right recursive path with deg(
c
) = deg(
a
) = 0,
c
<
a
; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
3
c
2
⇒
c
a
2
ca
[6]
2.
a
3
ca
⇒
c
[4]
3.
bc
⇒
c
a
2
ca
[5]
4.
ba
⇒
c
[2]
# ab:aaabaa=ba ca/b ba=c magic:0 aaacc=caaca aaaca=c bc=caaca ba=c