#260 ⟨
a
,
b
|
abab
=
aab
⟩
Up:
Monoid enumeration
Prev:
#259
⟨
a
,
b
|
abab
=
aaa
⟩
Next:
#261
⟨
a
,
b
|
abab
=
aba
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
bab
=
a
2
b
and
a
⋅
ab
=
a
2
b
, however
bab
≠
ab
Not right cancellative, because right multiplication by
b
is not injective:
aba
⋅
b
=
a
2
b
and
a
2
⋅
b
=
a
2
b
, however
aba
≠
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
Auxiliary generators:
c
=
aab
Reduction order:
Left-to-right recursive path with deg(
b
) = 0; deg(
c
) = deg(
a
) = 1,
c
<
a
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ac
⇒
cab
[5]
2.
abc
⇒
cab
[4]
3.
a
2
b
⇒
c
[2]
4.
(
ab
)
2
⇒
c
[3]
# ab:abab=aab b/ca aab=c magic:0 ac=cab abc=cab aab=c abab=c