#20158 ⟨
a
,
b
|
aab
=
b
,
bbab
=
bba
⟩
Up:
Monoid enumeration
Prev:
#20157
⟨
a
,
b
|
aab
=
b
,
bbab
=
bab
⟩
Next:
#20159
⟨
a
,
b
|
aab
=
b
,
bbab
=
bbb
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅
bab
=
b
2
a
and
b
⋅
ba
=
b
2
a
, however
bab
≠
ba
Not right cancellative, because right multiplication by
b
is not injective:
a
2
⋅
b
=
b
and 1 ⋅
b
=
b
, 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 recursive path with deg(
a
) = 0; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
2
b
⇒
b
[1]
2.
b
2
a
3
⇒
b
2
a
[4]
3.
b
3
⇒
b
2
a
2
[3]
4.
b
2
ab
⇒
b
2
a
[2]
# ab:aab=b,bbab=bba a/b - - aab=b bbaaa=bba bbb=bbaa bbab=bba