#3253 ⟨
a
,
b
,
c
|
bb
=
aa
,
acac
=1⟩
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Monoid enumeration
Prev:
#3252
⟨
a
,
b
,
c
|
bb
=
aa
,
abcc
=1⟩
Next:
#3254
⟨
a
,
b
,
c
|
bb
=
aa
,
acbc
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅ (
ca
)
2
=
a
and
a
⋅ 1 =
a
, however (
ca
)
2
≠ 1
Not right cancellative, because right multiplication by
cac
is not injective:
(
ca
)
2
⋅
cac
=
cac
and 1 ⋅
cac
=
cac
, however (
ca
)
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(
a
) = deg(
c
) = 0,
a
<
c
; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
(
ac
)
2
⇒ 1
[2]
2.
a
2
b
⇒
b
a
2
[3]
3.
b
2
⇒
a
2
[1]
# abc:bb=aa,acac=1 ac/b - - acac=1 aab=baa bb=aa