#1922 ⟨
a
,
b
,
c
|
abc
=
aa
,
cab
=1⟩
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Monoid enumeration
Prev:
#1920
⟨
a
,
b
,
c
|
abc
=
aa
,
bcc
=1⟩
Next:
#1925
⟨
a
,
b
,
c
|
abc
=
aa
,
cbb
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
ab
has infinite order
Not left cancellative, because left multiplication by
c
is not injective:
c
⋅ (
bc
)
2
=
c
and
c
⋅ 1 =
c
, however (
bc
)
2
≠ 1
Not right cancellative, because right multiplication by
bcb
is not injective:
(
bc
)
2
⋅
bcb
=
bcb
and 1 ⋅
bcb
=
bcb
, however (
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.
(
cb
)
2
⇒ 1
[5]
2.
ca
⇒
cbc
[4]
3.
a
2
⇒
abc
[1]
# abc:abc=aa,cab=1 bc/a - - cbcb=1 ca=cbc aa=abc