#17667 ⟨
a
,
b
|
aaaa
=1,
abbbb
=
bb
⟩
Up:
Monoid enumeration
Prev:
#17666
⟨
a
,
b
|
aaaa
=1,
abbbb
=
ba
⟩
Next:
#17670
⟨
a
,
b
|
aaaa
=1,
baaab
=
bb
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
a
2
ba
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅
b
9
=
b
2
and
b
⋅
b
=
b
2
, however
b
9
≠
b
Not right cancellative, because right multiplication by
b
is not injective:
b
9
⋅
b
=
b
2
and
b
⋅
b
=
b
2
, however
b
9
≠
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
10
⇒
b
2
[6]
2.
a
b
2
⇒
b
8
[5]
3.
a
4
⇒ 1
[1]
# ab:aaaa=1,abbbb=bb b/a - - bbbbbbbbbb=bb abb=bbbbbbbb aaaa=1