#2884 ⟨
a
,
b
,
c
|
aab
=
a
,
bcb
=
a
⟩
Up:
Monoid enumeration
Prev:
#2883
⟨
a
,
b
,
c
|
aab
=
a
,
bca
=
c
⟩
Next:
#2885
⟨
a
,
b
,
c
|
aab
=
a
,
bcb
=
b
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
b
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅ (
c
b
2
)
2
=
bcb
and
b
⋅
cb
=
bcb
, however (
c
b
2
)
2
≠
cb
Not right cancellative, because right multiplication by
b
is not injective:
(
bcb
)
2
⋅
b
=
bcb
and
bc
⋅
b
=
bcb
, however (
bcb
)
2
≠
bc
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(
b
) = deg(
c
) = 0,
b
<
c
; deg(
a
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
(
bc
)
2
b
2
⇒ (
bcb
)
2
[5]
2.
b
(
c
b
2
)
2
⇒
bcb
[4]
3.
a
⇒
bcb
[2]
# abc:aab=a,bcb=a bc/a - - bcbcbb=bcbbcb bcbbcbb=bcb a=bcb