#2309 ⟨
a
,
b
,
c
|
aaa
=
a
,
bbcb
=1⟩
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Monoid enumeration
Prev:
#2308
⟨
a
,
b
,
c
|
aaa
=
a
,
bbbc
=1⟩
Next:
#2310
⟨
a
,
b
,
c
|
aaa
=
a
,
bbcc
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Isomorphic to ℕ
(3 = 1)
∗ ℤ
Infinite non-cancellative non-commutative monoid
Element
aba
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
a
2
=
a
and
a
⋅ 1 =
a
, however
a
2
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
a
2
⋅
a
=
a
and 1 ⋅
a
=
a
, however
a
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.
cb
⇒
bc
[4]
2.
a
3
⇒
a
[1]
3.
b
3
c
⇒ 1
[5]
# abc:aaa=a,bbcb=1 abc - - cb=bc aaa=a bbbc=1