#2564 ⟨
a
,
b
|
abaaab
=
aaab
⟩
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Monoid enumeration
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#2563
⟨
a
,
b
|
abaaab
=
aaaa
⟩
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#2565
⟨
a
,
b
|
abaaab
=
aaba
⟩
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
⋅
b
a
3
b
=
a
3
b
and
a
⋅
a
2
b
=
a
3
b
, however
b
a
3
b
≠
a
2
b
Not right cancellative, because right multiplication by
c
is not injective:
ab
⋅
c
=
c
and 1 ⋅
c
=
c
, however
ab
≠ 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
Auxiliary generators:
c
=
aaab
Reduction order:
Left-to-right shortlex with
a
<
b
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
2
c
⇒
c
2
[5]
2.
abc
⇒
c
[4]
3.
a
3
b
⇒
c
[2]
# ab:abaaab=aaab abc aaab=c magic:0 aac=cc abc=c aaab=c