#4151 ⟨
a
,
b
|
abb
=
aaa
,
baa
=
b
⟩
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Monoid enumeration
Prev:
#4150
⟨
a
,
b
|
abb
=
aaa
,
baa
=
a
⟩
Next:
#4152
⟨
a
,
b
|
abb
=
aaa
,
bab
=
a
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element
ab
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅
a
2
=
b
and
b
⋅ 1 =
b
, however
a
2
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
b
2
⋅
b
=
b
and 1 ⋅
b
=
b
, however
b
2
≠ 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
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
b
2
⇒
a
3
[1]
2.
b
a
2
⇒
b
[2]
3.
b
3
⇒
b
[4]
4.
a
3
b
⇒
ab
[5]
5.
a
5
⇒
a
3
[3]
# ab:abb=aaa,baa=b ab - - abb=aaa baa=b bbb=b aaab=ab aaaaa=aaa