#6915 ⟨
a
,
b
,
c
|
ab
=1,
acaca
=
c
⟩
Up:
Monoid enumeration
Prev:
#6906
⟨
a
,
b
,
c
|
ab
=1,
acaac
=
c
⟩
Next:
#6916
⟨
a
,
b
,
c
|
ab
=1,
acacb
=
a
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
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
and
a
⋅ 1 =
a
, however
ba
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
ba
⋅
b
=
b
and 1 ⋅
b
=
b
, however
ba
≠ 1
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
) = deg(
c
) = 0,
a
<
c
; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
c
2
a
⇒
a
c
2
[4]
2.
a
(
ca
)
2
⇒
c
[2]
3.
ab
⇒ 1
[1]
4.
cb
⇒ (
ac
)
2
[3]
# abc:ab=1,acaca=c ac/b - - cca=acc acaca=c ab=1 cb=acac