#3030 ⟨
a
,
b
,
c
|
aba
=
b
,
aca
=
b
⟩
Up:
Monoid enumeration
Prev:
#3029
⟨
a
,
b
,
c
|
aba
=
b
,
abc
=
c
⟩
Next:
#3031
⟨
a
,
b
,
c
|
aba
=
b
,
aca
=
c
⟩
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
⋅
ac
a
2
=
aca
and
a
⋅
ca
=
aca
, however
ac
a
2
≠
ca
Not right cancellative, because right multiplication by
a
is not injective:
a
2
ca
⋅
a
=
aca
and
ac
⋅
a
=
aca
, however
a
2
ca
≠
ac
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.
a
2
c
a
2
⇒
aca
[4]
2.
(
ac
)
2
a
2
⇒
a
2
(
ca
)
2
[5]
3.
b
⇒
aca
[2]
# abc:aba=b,aca=b ac/b - - aacaa=aca acacaa=aacaca b=aca