#4137 ⟨
a
,
b
|
aba
=
aaa
,
bab
=
b
⟩
Up:
Monoid enumeration
Prev:
#4136
⟨
a
,
b
|
aba
=
aaa
,
bab
=
a
⟩
Next:
#4139
⟨
a
,
b
|
aba
=
aaa
,
bbb
=
b
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element
b
2
a
2
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅
ab
=
b
and
b
⋅ 1 =
b
, however
ab
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
ba
⋅
b
=
b
and 1 ⋅
b
=
b
, however
ba
≠ 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.
bab
⇒
b
[2]
2.
a
3
⇒
aba
[1]
3.
a
2
ba
⇒
ab
a
2
[3]
4.
ab
a
2
b
⇒
a
2
b
[4]
# ab:aba=aaa,bab=b ba - - bab=b aaa=aba aaba=abaa abaab=aab