#4425 ⟨
a
,
b
|
aaaabaaa
=
aba
⟩
Up:
Monoid enumeration
Prev:
#4424
⟨
a
,
b
|
aaaabaaa
=
aab
⟩
Next:
#4426
⟨
a
,
b
|
aaaabaaa
=
abb
⟩
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
⋅
a
3
b
a
3
=
aba
and
a
⋅
ba
=
aba
, however
a
3
b
a
3
≠
ba
Not right cancellative, because right multiplication by
a
is not injective:
a
4
b
a
2
⋅
a
=
aba
and
ab
⋅
a
=
aba
, however
a
4
b
a
2
≠
ab
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
4
(
ba
)
2
⇒
ab
a
2
b
a
3
[2]
2.
a
4
b
a
3
⇒
aba
[1]
3.
a
3
(
aba
)
2
⇒
a
(
b
a
3
)
2
[3]
# ab:aaaabaaa=aba ba - - aaaababa=abaabaaa aaaabaaa=aba aaaabaaba=abaaabaaa