#502 ⟨
a
,
b
,
c
|
bb
=
ac
,
cba
=1⟩
Up:
Monoid enumeration
Prev:
#501
⟨
a
,
b
,
c
|
bb
=
ac
,
caa
=1⟩
Next:
#503
⟨
a
,
b
,
c
|
ab
=
aa
,
ac
=
a
⟩
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
c
is not injective:
c
⋅ (
ac
)
3
=
c
and
c
⋅ 1 =
c
, however (
ac
)
3
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
b
⋅
a
=
a
(
ca
)
2
and (
ac
)
2
⋅
a
=
a
(
ca
)
2
, however
b
≠ (
ac
)
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
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.
(
ca
)
3
⇒ 1
[7]
2.
ba
⇒
a
(
ca
)
2
[6]
3.
cb
⇒
c
(
ac
)
2
[4]
4.
b
2
⇒
ac
[1]
# abc:bb=ac,cba=1 ac/b - - cacaca=1 ba=acaca cb=cacac bb=ac