#7453 ⟨
a
,
b
,
c
|
ab
=1,
accc
=
bb
⟩
Up:
Monoid enumeration
Prev:
#7452
⟨
a
,
b
,
c
|
ab
=1,
accc
=
ba
⟩
Next:
#7454
⟨
a
,
b
,
c
|
ab
=1,
accc
=
bc
⟩
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
2
c
3
a
=
a
and
a
⋅ 1 =
a
, however
a
2
c
3
a
≠ 1
Not right cancellative, because right multiplication by
a
2
c
3
is not injective:
a
2
c
3
a
⋅
a
2
c
3
=
a
2
c
3
and 1 ⋅
a
2
c
3
=
a
2
c
3
, however
a
2
c
3
a
≠ 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(
a
) = deg(
c
) = 0,
a
<
c
; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
3
c
3
⇒ 1
[5]
2.
a
c
3
a
2
c
3
⇒
a
(
a
c
3
)
2
[4]
3.
b
⇒
a
2
c
3
[3]
# abc:ab=1,accc=bb ac/b - - aaaccc=1 acccaaccc=aacccaccc b=aaccc