#4703 ⟨
a
,
b
|
aaba
=
a
,
abab
=
b
⟩
Up:
Monoid enumeration
Prev:
#4700
⟨
a
,
b
|
aaba
=
a
,
abaa
=
a
⟩
Next:
#4704
⟨
a
,
b
|
aaba
=
a
,
abba
=
a
⟩
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
⋅
aba
=
a
and
a
⋅ 1 =
a
, however
aba
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
a
2
b
⋅
a
=
a
and 1 ⋅
a
=
a
, however
a
2
b
≠ 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.
bab
⇒
a
b
2
[4]
2.
a
2
ba
⇒
a
[1]
3.
a
2
b
2
⇒
b
[3]
# ab:aaba=a,abab=b ab - - bab=abb aaba=a aabb=b