#22856 ⟨
a
,
b
|
aaa
=1,
bbabb
=
bab
⟩
Up:
Monoid enumeration
Prev:
#22855
⟨
a
,
b
|
aaa
=1,
bbabb
=
abb
⟩
Next:
#22857
⟨
a
,
b
|
aaa
=1,
bbabb
=
bbb
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
aba
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅
ba
b
2
=
bab
and
b
⋅
ab
=
bab
, however
ba
b
2
≠
ab
Not right cancellative, because right multiplication by
b
is not injective:
b
2
ab
⋅
b
=
bab
and
ba
⋅
b
=
bab
, however
b
2
ab
≠
ba
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
3
⇒ 1
[1]
2.
b
2
a
b
2
⇒
bab
[2]
3.
b
2
(
ab
)
2
⇒ (
ba
)
2
b
2
[3]
# ab:aaa=1,bbabb=bab ab - - aaa=1 bbabb=bab bbabab=bababb