#16011 ⟨
a
,
b
|
aaa
=
ab
,
abbb
=
ba
⟩
Up:
Monoid enumeration
Prev:
#16010
⟨
a
,
b
|
aaa
=
ab
,
abbb
=
ab
⟩
Next:
#16012
⟨
a
,
b
|
aaa
=
ab
,
abbb
=
bb
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
b
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
b
=
a
3
and
a
⋅
a
2
=
a
3
, however
b
≠
a
2
Not right cancellative, because right multiplication by
a
is not injective:
a
7
⋅
a
=
a
4
and
a
3
⋅
a
=
a
4
, however
a
7
≠
a
3
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 recursive path with deg(
a
) = 0; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
8
⇒
a
4
[3]
2.
ba
⇒
a
7
[2]
3.
ab
⇒
a
3
[1]
# ab:aaa=ab,abbb=ba a/b - - aaaaaaaa=aaaa ba=aaaaaaa ab=aaa