#3066 ⟨
a
,
b
,
c
|
abc
=
a
,
cab
=
b
⟩
Up:
Monoid enumeration
Prev:
#3065
⟨
a
,
b
,
c
|
abc
=
a
,
cab
=
a
⟩
Next:
#3067
⟨
a
,
b
,
c
|
abc
=
a
,
cba
=
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
⋅
bc
=
a
and
a
⋅ 1 =
a
, however
bc
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
bc
⋅
b
=
b
and 1 ⋅
b
=
b
, however
bc
≠ 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
a
<
b
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ca
⇒
bc
[4]
2.
a
b
2
⇒
a
2
b
[3]
3.
abc
⇒
a
[1]
4.
bcb
⇒
b
[5]
# abc:abc=a,cab=b abc - - ca=bc abb=aab abc=a bcb=b