#5820 ⟨
a
,
b
,
c
|
ab
=
a
,
aca
=
ca
⟩
Up:
Monoid enumeration
Prev:
#5819
⟨
a
,
b
,
c
|
ab
=
a
,
aca
=
bc
⟩
Next:
#5821
⟨
a
,
b
,
c
|
ab
=
a
,
aca
=
cb
⟩
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
⋅
b
=
a
and
a
⋅ 1 =
a
, however
b
≠ 1
Not right cancellative, because right multiplication by
d
is not injective:
a
⋅
d
=
d
and 1 ⋅
d
=
d
, however
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
Auxiliary generators:
d
=
ca
Reduction order:
Left-to-right shortlex with
a
<
b
<
d
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
Morphocompletion
: how the auxiliary generators were found.
#
Rule
Proof
1.
ab
⇒
a
[1]
2.
ad
⇒
d
[5]
3.
db
⇒
d
[6]
4.
ca
⇒
d
[3]
5.
cd
⇒
d
2
[7]
# abc:ab=a,aca=ca abdc ca=d morph:2/1 ab=a ad=d db=d ca=d cd=dd