#22822 ⟨
a
,
b
|
aaa
=1,
abbbb
=
bba
⟩
Up:
Monoid enumeration
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#22821
⟨
a
,
b
|
aaa
=1,
abbbb
=
bab
⟩
Next:
#22823
⟨
a
,
b
|
aaa
=1,
abbbb
=
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
⋅
b
15
=
b
2
and
b
⋅
b
=
b
2
, however
b
15
≠
b
Not right cancellative, because right multiplication by
b
is not injective:
b
15
⋅
b
=
b
2
and
b
⋅
b
=
b
2
, however
b
15
≠
b
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
16
⇒
b
2
[3]
2.
b
2
a
⇒
a
b
4
[2]
3.
a
3
⇒ 1
[1]
# ab:aaa=1,abbbb=bba b/a - - bbbbbbbbbbbbbbbb=bb bba=abbbb aaa=1