#22843 ⟨
a
,
b
|
aaa
=1,
babab
=
bbb
⟩
Up:
Monoid enumeration
Prev:
#22842
⟨
a
,
b
|
aaa
=1,
babab
=
bab
⟩
Next:
#22847
⟨
a
,
b
|
aaa
=1,
babbb
=
abb
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
b
a
2
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅ (
ab
)
2
=
b
3
and
b
⋅
b
2
=
b
3
, however (
ab
)
2
≠
b
2
Not right cancellative, because right multiplication by
b
is not injective:
(
ba
)
2
⋅
b
=
b
3
and
b
2
⋅
b
=
b
3
, however (
ba
)
2
≠
b
2
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
(
ab
)
2
⇒
b
3
[2]
3.
b
3
ab
⇒
ba
b
3
[3]
# ab:aaa=1,babab=bbb ab - - aaa=1 babab=bbb bbbab=babbb