#5733 ⟨
a
,
b
|
aaaa
=1,
bbbbb
=
b
⟩
Up:
Monoid enumeration
Prev:
#5731
⟨
a
,
b
|
aaaa
=1,
bbabb
=
b
⟩
Next:
#5736
⟨
a
,
b
|
aaab
=1,
aaaab
=
a
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 10
Isomorphic to ℤ
4
∗ ℕ
(5 = 1)
Infinite non-cancellative non-commutative monoid
Element
a
2
ba
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅
b
4
=
b
and
b
⋅ 1 =
b
, however
b
4
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
b
4
⋅
b
=
b
and 1 ⋅
b
=
b
, however
b
4
≠ 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
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
4
⇒ 1
[1]
2.
b
5
⇒
b
[2]
# ab:aaaa=1,bbbbb=b ab - - aaaa=1 bbbbb=b