#12944 ⟨
a
,
b
|
aba
=
aab
,
abba
=
a
⟩
Up:
Monoid enumeration
Prev:
#12932
⟨
a
,
b
|
aba
=
aab
,
aaaa
=
a
⟩
Next:
#12946
⟨
a
,
b
|
aba
=
aab
,
abbb
=
a
⟩
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
⋅
b
2
a
=
a
and
a
⋅ 1 =
a
, however
b
2
a
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
a
b
2
⋅
a
=
a
and 1 ⋅
a
=
a
, however
a
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
b
<
a
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
2
b
⇒
aba
[1]
2.
a
b
2
a
⇒
a
[2]
3.
a
(
ba
)
2
⇒
a
2
[3]
# ab:aba=aab,abba=a ba - - aab=aba abba=a ababa=aa