#1539 ⟨
a
,
b
,
c
|
ab
=1,
ccc
=
aa
⟩
Up:
Monoid enumeration
Prev:
#1538
⟨
a
,
b
,
c
|
ab
=1,
cac
=
cc
⟩
Next:
#1541
⟨
a
,
b
,
c
|
ab
=1,
ccc
=
ac
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element
b
has infinite order
Not left cancellative, because left multiplication by
c
is not injective:
c
⋅
c
2
b
2
c
=
c
and
c
⋅ 1 =
c
, however
c
2
b
2
c
≠ 1
Not right cancellative, because right multiplication by
c
2
b
2
is not injective:
c
2
b
2
c
⋅
c
2
b
2
=
c
2
b
2
and 1 ⋅
c
2
b
2
=
c
2
b
2
, however
c
2
b
2
c
≠ 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(
b
) = deg(
c
) = 0,
b
<
c
; deg(
a
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
c
3
b
2
⇒ 1
[5]
2.
c
6
b
⇒
c
3
b
c
3
[4]
3.
(
c
3
b
)
2
⇒
c
3
[6]
4.
a
⇒
c
3
b
[3]
# abc:ab=1,ccc=aa bc/a - - cccbb=1 ccccccb=cccbccc cccbcccb=ccc a=cccb