#20142 ⟨
a
,
b
|
aab
=
b
,
babb
=
bba
⟩
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Monoid enumeration
Prev:
#20141
⟨
a
,
b
|
aab
=
b
,
babb
=
bab
⟩
Next:
#20143
⟨
a
,
b
|
aab
=
b
,
babb
=
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
⋅
ba
=
ba
b
2
and
b
⋅
a
b
2
=
ba
b
2
, however
ba
≠
a
b
2
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(
b
) = 0; deg(
a
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
b
7
⇒
b
4
[4]
2.
b
2
a
⇒
ba
b
2
[2]
3.
a
2
b
⇒
b
[1]
4.
(
ba
)
2
b
3
⇒
b
3
[3]
# ab:aab=b,babb=bba b/a - - bbbbbbb=bbbb bba=babb aab=b bababbb=bbb