#20234 ⟨
a
,
b
|
aba
=
a
,
babb
=
aaa
⟩
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Monoid enumeration
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#20233
⟨
a
,
b
|
aba
=
a
,
baab
=
bbb
⟩
Next:
#20235
⟨
a
,
b
|
aba
=
a
,
babb
=
aab
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
b
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
ba
=
a
and
a
⋅ 1 =
a
, however
ba
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
ab
⋅
a
=
a
and 1 ⋅
a
=
a
, 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
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.
a
8
⇒
a
3
[5]
2.
b
a
3
⇒
a
7
[4]
3.
aba
⇒
a
[1]
4.
a
b
2
⇒
a
4
[3]
# ab:aba=a,babb=aaa a/b - - aaaaaaaa=aaa baaa=aaaaaaa aba=a abb=aaaa