#4123 ⟨
a
,
b
,
c
|
aaa
=1,
abcb
=
b
⟩
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Monoid enumeration
Prev:
#4121
⟨
a
,
b
,
c
|
aaa
=1,
abca
=
b
⟩
Next:
#4124
⟨
a
,
b
,
c
|
aaa
=1,
abcb
=
c
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
a
2
c
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅ (
cb
)
3
=
b
and
b
⋅ 1 =
b
, however (
cb
)
3
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
(
bc
)
3
⋅
b
=
b
and 1 ⋅
b
=
b
, however (
bc
)
3
≠ 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.
b
(
cb
)
3
⇒
b
[5]
2.
ab
⇒
b
(
cb
)
2
[4]
3.
a
3
⇒ 1
[1]
# abc:aaa=1,abcb=b bc/a - - bcbcbcb=b ab=bcbcb aaa=1