#3263 ⟨
a
,
b
,
c
|
bb
=
ac
,
aaab
=1⟩
Up:
Monoid enumeration
Prev:
#3261
⟨
a
,
b
,
c
|
bb
=
aa
,
cccc
=1⟩
Next:
#3264
⟨
a
,
b
,
c
|
bb
=
ac
,
aaac
=1⟩
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
⋅
a
6
ca
=
a
and
a
⋅ 1 =
a
, however
a
6
ca
≠ 1
Not right cancellative, because right multiplication by
c
is not injective:
ac
a
4
⋅
c
=
a
4
cac
and
a
4
ca
⋅
c
=
a
4
cac
, however
ac
a
4
≠
a
4
ca
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
4
c
⇒
a
4
cac
[5]
2.
a
7
c
⇒ 1
[6]
3.
b
⇒
a
4
c
[4]
# abc:bb=ac,aaab=1 reversed:ca/b - - acaaaac=aaaacac aaaaaaac=1 b=aaaac