#6792 ⟨
a
,
b
,
c
|
ab
=1,
aaaca
=
c
⟩
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Monoid enumeration
Prev:
#6784
⟨
a
,
b
,
c
|
ab
=1,
aaaac
=
c
⟩
Next:
#6793
⟨
a
,
b
,
c
|
ab
=1,
aaacb
=
a
⟩
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
⋅
ba
=
a
and
a
⋅ 1 =
a
, however
ba
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
ba
⋅
b
=
b
and 1 ⋅
b
=
b
, however
ba
≠ 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(
c
) = deg(
a
) = 0,
c
<
a
; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
3
c
2
⇒
c
a
2
ca
[4]
2.
a
3
ca
⇒
c
[2]
3.
cb
⇒
a
3
c
[3]
4.
ab
⇒ 1
[1]
# abc:ab=1,aaaca=c ca/b - - aaacc=caaca aaaca=c cb=aaac ab=1