#1876 ⟨
a
,
b
,
c
|
aba
=
bc
,
cab
=1⟩
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Monoid enumeration
Prev:
#1874
⟨
a
,
b
,
c
|
aba
=
bc
,
bcc
=1⟩
Next:
#1877
⟨
a
,
b
,
c
|
aba
=
bc
,
cac
=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
c
is not injective:
c
⋅
c
(
bc
)
2
=
c
and
c
⋅ 1 =
c
, however
c
(
bc
)
2
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
c
(
bc
)
2
⋅
b
=
b
and 1 ⋅
b
=
b
, however
c
(
bc
)
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.
c
b
2
⇒
bcb
[7]
2.
c
(
cb
)
2
⇒ 1
[5]
3.
(
cb
)
3
⇒
b
[6]
4.
a
⇒
cbc
[3]
# abc:aba=bc,cab=1 bc/a - - cbb=bcb ccbcb=1 cbcbcb=b a=cbc