#3646 ⟨
a
,
b
,
c
|
aaa
=1,
bbcbc
=1⟩
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Monoid enumeration
Prev:
#3644
⟨
a
,
b
,
c
|
aaa
=1,
bbbcc
=1⟩
Next:
#3647
⟨
a
,
b
,
c
|
aaa
=1,
bbccb
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
aba
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅
b
(
cb
)
2
=
b
and
b
⋅ 1 =
b
, however
b
(
cb
)
2
≠ 1
Not right cancellative, because right multiplication by (
bc
)
2
is not injective:
b
(
cb
)
2
⋅ (
bc
)
2
= (
bc
)
2
and 1 ⋅ (
bc
)
2
= (
bc
)
2
, however
b
(
cb
)
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 shortlex with
a
<
b
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
3
⇒ 1
[1]
2.
b
(
bc
)
2
⇒ 1
[2]
# abc:aaa=1,bbcbc=1 abc - - aaa=1 bbcbc=1