#4501 ⟨
a
,
b
,
c
|
abc
=1,
aaca
=
c
⟩
Up:
Monoid enumeration
Prev:
#4496
⟨
a
,
b
,
c
|
abc
=1,
aabc
=
a
⟩
Next:
#4502
⟨
a
,
b
,
c
|
abc
=1,
aacb
=
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
b
is not injective:
b
⋅
cab
=
b
and
b
⋅ 1 =
b
, however
cab
≠ 1
Not right cancellative, because right multiplication by
c
is not injective:
cab
⋅
c
=
c
and 1 ⋅
c
=
c
, however
cab
≠ 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 shortlex with
c
<
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
2
c
⇒
cbc
[3]
2.
abc
⇒ 1
[1]
3.
b
c
2
⇒
aca
[7]
4.
bca
⇒ 1
[5]
# abc:abc=1,aaca=c cab - - aac=cbc abc=1 bcc=aca bca=1