#20215 ⟨
a
,
b
|
aba
=
a
,
abba
=
aab
⟩
Up:
Monoid enumeration
Prev:
#20214
⟨
a
,
b
|
aba
=
a
,
abba
=
aaa
⟩
Next:
#20217
⟨
a
,
b
|
aba
=
a
,
abba
=
abb
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
a
2
b
4
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 shortlex with
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
3
⇒
a
2
[5]
2.
aba
⇒
a
[1]
3.
a
b
2
a
⇒
a
2
b
[2]
4.
a
2
b
3
a
⇒
a
2
b
[4]
# ab:aba=a,abba=aab ab - - aaa=aa aba=a abba=aab aabbba=aab