#1536 ⟨
a
,
b
,
c
|
ab
=1,
cac
=
ca
⟩
Up:
Monoid enumeration
Prev:
#1535
⟨
a
,
b
,
c
|
ab
=1,
cac
=
bc
⟩
Next:
#1537
⟨
a
,
b
,
c
|
ab
=1,
cac
=
cb
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
d
is not injective:
d
⋅
db
=
d
and
d
⋅ 1 =
d
, however
db
≠ 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
Auxiliary generators:
d
=
ca
Reduction order:
Left-to-right recursive path with deg(
b
) = deg(
d
) = 0,
b
<
d
; deg(
a
) = deg(
c
) = 1,
a
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
Morphocompletion
: how the auxiliary generators were found.
#
Rule
Proof
1.
d
2
b
⇒
d
[9]
2.
ab
⇒ 1
[1]
3.
da
⇒
d
2
[7]
4.
dba
⇒
d
[8]
5.
c
⇒
db
[6]
# abc:ab=1,cac=ca bd/ac ca=d morph:2/1 ddb=d ab=1 da=dd dba=d c=db