#5900 ⟨
a
,
b
|
abab
=1,
aaabb
=
a
⟩
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Monoid enumeration
Prev:
#5878
⟨
a
,
b
|
aabb
=1,
abaab
=
a
⟩
Next:
#5901
⟨
a
,
b
|
abab
=1,
aaabb
=
b
⟩
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
⋅ (
ba
)
2
=
a
and
a
⋅ 1 =
a
, however (
ba
)
2
≠ 1
Not right cancellative, because right multiplication by
bab
is not injective:
(
ba
)
2
⋅
bab
=
bab
and 1 ⋅
bab
=
bab
, however (
ba
)
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.
(
ab
)
2
⇒ 1
[1]
2.
a
3
b
2
⇒
a
[2]
# ab:abab=1,aaabb=a ab - - abab=1 aaabb=a