#17117 ⟨
a
,
b
|
aaaa
=1,
aabbbb
=
b
⟩
Up:
Monoid enumeration
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#17113
⟨
a
,
b
|
aaaa
=1,
aabbab
=
b
⟩
Next:
#17135
⟨
a
,
b
|
aaaa
=1,
abbbab
=
b
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
b
a
3
has infinite order
Not right cancellative, because right multiplication by
b
is not injective:
b
3
⋅
b
=
a
2
b
and
a
2
⋅
b
=
a
2
b
, however
b
3
≠
a
2
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 shortlex with
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
4
⇒ 1
[1]
2.
b
a
2
b
⇒
a
2
b
2
[4]
3.
b
4
⇒
a
2
b
[3]
# ab:aaaa=1,aabbbb=b ab - - aaaa=1 baab=aabb bbbb=aab