#489 ⟨
a
,
b
,
c
|
bb
=
ac
,
aab
=1⟩
Up:
Monoid enumeration
Prev:
#487
⟨
a
,
b
,
c
|
bb
=
aa
,
ccc
=1⟩
Next:
#490
⟨
a
,
b
,
c
|
bb
=
ac
,
aac
=1⟩
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
⋅
a
4
ca
=
a
and
a
⋅ 1 =
a
, however
a
4
ca
≠ 1
Not right cancellative, because right multiplication by
a
4
c
is not injective:
a
4
ca
⋅
a
4
c
=
a
4
c
and 1 ⋅
a
4
c
=
a
4
c
, however
a
4
ca
≠ 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:
Right-to-left recursive path with deg(
c
) = deg(
a
) = 0,
c
<
a
; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ac
a
3
c
⇒
a
3
cac
[5]
2.
a
5
c
⇒ 1
[6]
3.
b
⇒
a
3
c
[4]
# abc:bb=ac,aab=1 reversed:ca/b - - acaaac=aaacac aaaaac=1 b=aaac