#6185 ⟨
a
,
b
,
c
|
aa
=1,
abcbcc
=1⟩
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Monoid enumeration
Prev:
#6180
⟨
a
,
b
,
c
|
aa
=1,
abcbac
=1⟩
Next:
#6186
⟨
a
,
b
,
c
|
aa
=1,
abccab
=1⟩
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
⋅ (
cb
c
2
b
)
2
=
b
and
b
⋅ 1 =
b
, however (
cb
c
2
b
)
2
≠ 1
Not right cancellative, because right multiplication by (
cbc
)
2
b
c
2
is not injective:
(
cb
c
2
b
)
2
⋅ (
cbc
)
2
b
c
2
= (
cbc
)
2
b
c
2
and 1 ⋅ (
cbc
)
2
b
c
2
= (
cbc
)
2
b
c
2
, however (
cb
c
2
b
)
2
≠ 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(
b
) = deg(
c
) = 0,
b
<
c
; deg(
a
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
(
bcb
c
2
)
2
⇒ 1
[4]
2.
a
⇒
bcb
c
2
[3]
# abc:aa=1,abcbcc=1 bc/a - - bcbccbcbcc=1 a=bcbcc