#1843 ⟨
a
,
b
,
c
|
aba
=
ac
,
cbb
=1⟩
Up:
Monoid enumeration
Prev:
#1838
⟨
a
,
b
,
c
|
aba
=
ac
,
bcc
=1⟩
Next:
#1844
⟨
a
,
b
,
c
|
aba
=
ac
,
cbc
=1⟩
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
c
is not injective:
c
⋅
b
2
c
=
c
and
c
⋅ 1 =
c
, however
b
2
c
≠ 1
Not right cancellative, because right multiplication by
b
2
is not injective:
b
2
c
⋅
b
2
=
b
2
and 1 ⋅
b
2
=
b
2
, however
b
2
c
≠ 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 recursive path with deg(
a
) = deg(
b
) = 0,
a
<
b
; deg(
c
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
aba
b
2
⇒
a
[3]
2.
c
b
2
⇒ 1
[2]
3.
ac
⇒
aba
[1]
# abc:aba=ac,cbb=1 ab/c - - ababb=a cbb=1 ac=aba