#7600 ⟨
a
,
b
,
c
|
ab
=1,
cbac
=
ac
⟩
Up:
Monoid enumeration
Prev:
#7598
⟨
a
,
b
,
c
|
ab
=1,
cbac
=
aa
⟩
Next:
#7601
⟨
a
,
b
,
c
|
ab
=1,
cbac
=
ba
⟩
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
⋅
ba
=
a
and
a
⋅ 1 =
a
, however
ba
≠ 1
Not right cancellative, because right multiplication by
d
is not injective:
cb
⋅
d
=
d
and 1 ⋅
d
=
d
, however
cb
≠ 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
=
ac
Reduction order:
Left-to-right recursive path with deg(
b
) = deg(
c
) = deg(
d
) = 0,
b
<
c
<
d
; deg(
a
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
Morphocompletion
: how the auxiliary generators were found.
#
Rule
Proof
1.
cbd
⇒
d
[5]
2.
ab
⇒ 1
[1]
3.
ac
⇒
d
[3]
4.
ad
⇒
dbd
[6]
# abc:ab=1,cbac=ac bcd/a ac=d morph:2/1 cbd=d ab=1 ac=d ad=dbd