#3032 ⟨
a
,
b
,
c
|
aba
=
b
,
acb
=
a
⟩
Up:
Monoid enumeration
Prev:
#3031
⟨
a
,
b
,
c
|
aba
=
b
,
aca
=
c
⟩
Next:
#3033
⟨
a
,
b
,
c
|
aba
=
b
,
acb
=
b
⟩
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
⋅
cb
=
b
and
b
⋅ 1 =
b
, however
cb
≠ 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
b
<
a
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
bcb
⇒
b
[4]
2.
a
b
2
⇒
b
2
a
[3]
3.
aba
⇒
b
[1]
4.
acb
⇒
a
[2]
# abc:aba=b,acb=a bac - - bcb=b abb=bba aba=b acb=a