#1106 ⟨
a
,
b
|
abaaab
=
abb
⟩
Up:
Monoid enumeration
Prev:
#1105
⟨
a
,
b
|
abaaab
=
aba
⟩
Next:
#1107
⟨
a
,
b
|
abaaab
=
baa
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 9
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
b
a
3
b
=
a
b
2
and
a
⋅
b
2
=
a
b
2
, however
b
a
3
b
≠
b
2
Not right cancellative, because right multiplication by
b
is not injective:
ab
a
3
⋅
b
=
a
b
2
and
ab
⋅
b
=
a
b
2
, however
ab
a
3
≠
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
Auxiliary generators:
c
=
abb
Reduction order:
Left-to-right recursive path with deg(
a
) = deg(
c
) = 0,
a
<
c
; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
cb
⇒
ab
a
2
c
[5]
2.
c
a
3
b
⇒
ab
a
2
c
[4]
3.
a
b
2
⇒
c
[2]
4.
ab
a
3
b
⇒
c
[3]
# ab:abaaab=abb ac/b abb=c magic:0 cb=abaac caaab=abaac abb=c abaaab=c