#4126 ⟨
a
,
b
|
aab
=
aaa
,
bab
=
a
⟩
Up:
Monoid enumeration
Prev:
#4125
⟨
a
,
b
|
aab
=
aaa
,
baa
=
b
⟩
Next:
#4127
⟨
a
,
b
|
aab
=
aaa
,
bab
=
b
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element
b
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
ab
=
a
3
and
a
⋅
a
2
=
a
3
, however
ab
≠
a
2
Not right cancellative, because right multiplication by
a
is not injective:
ba
⋅
a
=
a
3
and
a
2
⋅
a
=
a
3
, however
ba
≠
a
2
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
2
b
⇒
a
3
[1]
2.
b
a
2
⇒
a
3
[3]
3.
bab
⇒
a
[2]
4.
a
5
⇒
a
3
[4]
# ab:aab=aaa,bab=a ab - - aab=aaa baa=aaa bab=a aaaaa=aaa