#12311 ⟨
a
,
b
|
aaab
=
bb
,
baaa
=
b
⟩
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Monoid enumeration
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#12309
⟨
a
,
b
|
aaab
=
bb
,
abbb
=
b
⟩
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#12313
⟨
a
,
b
|
aaab
=
bb
,
baab
=
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
b
is not injective:
b
⋅
a
3
=
b
and
b
⋅ 1 =
b
, however
a
3
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
b
⋅
b
=
a
3
b
and
a
3
⋅
b
=
a
3
b
, however
b
≠
a
3
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 recursive path with deg(
a
) = 0; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
b
a
3
⇒
b
[2]
2.
a
6
b
⇒
a
3
b
[3]
3.
b
2
⇒
a
3
b
[1]
# ab:aaab=bb,baaa=b a/b - - baaa=b aaaaaab=aaab bb=aaab