#16439 ⟨
a
,
b
|
aba
=
bb
,
aaaa
=
bb
⟩
Up:
Monoid enumeration
Prev:
#16437
⟨
a
,
b
|
aba
=
bb
,
aaaa
=
aa
⟩
Next:
#16441
⟨
a
,
b
|
aba
=
bb
,
aaab
=
ab
⟩
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
⋅
ba
=
a
4
and
a
⋅
a
3
=
a
4
, however
ba
≠
a
3
Not right cancellative, because right multiplication by
a
is not injective:
ab
⋅
a
=
a
4
and
a
3
⋅
a
=
a
4
, however
ab
≠
a
3
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 recursive path with deg(
a
) = 0; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
b
a
5
⇒
a
7
[5]
2.
aba
⇒
a
4
[2]
3.
a
4
b
⇒
b
a
4
[4]
4.
b
2
⇒
a
4
[3]
# ab:aba=bb,aaaa=bb a/b - - baaaaa=aaaaaaa aba=aaaa aaaab=baaaa bb=aaaa