#18023 ⟨
a
,
b
|
abab
=1,
aaabb
=
bb
⟩
Up:
Monoid enumeration
Prev:
#18022
⟨
a
,
b
|
abab
=1,
aaabb
=
ba
⟩
Next:
#18034
⟨
a
,
b
|
abab
=1,
aabba
=
ba
⟩
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
a
is not injective:
a
⋅ (
ba
)
2
=
a
and
a
⋅ 1 =
a
, however (
ba
)
2
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
a
3
b
⋅
b
=
b
2
and
b
⋅
b
=
b
2
, however
a
3
b
≠
b
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
⇒
b
2
[2]
# ab:abab=1,aaabb=bb ab - - abab=1 aaabb=bb