#2590 ⟨
a
,
b
|
ababab
=
aaab
⟩
Up:
Monoid enumeration
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#2589
⟨
a
,
b
|
ababab
=
aaaa
⟩
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#2591
⟨
a
,
b
|
ababab
=
aaba
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
b
(
ab
)
2
=
a
3
b
and
a
⋅
a
2
b
=
a
3
b
, however
b
(
ab
)
2
≠
a
2
b
Not right cancellative, because right multiplication by
b
is not injective:
a
(
ba
)
2
⋅
b
=
a
3
b
and
a
3
⋅
b
=
a
3
b
, however
a
(
ba
)
2
≠
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
Auxiliary generators:
c
=
aaab
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.
abc
⇒
cab
[4]
2.
a
2
c
⇒
c
(
ab
)
2
[5]
3.
a
3
b
⇒
c
[2]
4.
(
ab
)
3
⇒
c
[3]
# ab:ababab=aaab b/ca aaab=c magic:0 abc=cab aac=cabab aaab=c ababab=c