#23 ⟨
a
,
b
|
bb
=
aa
⟩
Up:
Monoid enumeration
Prev:
#22
⟨
a
,
b
|
ba
=
ab
⟩
Next:
#24
⟨
a
,
b
|
aaaaa
=1⟩
Contents
Properties
Rewriting system
Other submonoids of same group
Properties
Sum of relation sides is 4
Infinite cancellative non-commutative monoid
Element
a
has infinite order
Submonoid of enveloping group: ⟨
a
,
b
|
aabb
⟩
Embedding:
φ
(
a
) =
a
-1
,
φ
(
b
) =
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 shortlex with
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
b
2
⇒
a
2
[1]
2.
b
a
2
⇒
a
2
b
[2]
# ab:bb=aa ab - - bb=aa baa=aab
Other submonoids of same group
2 unique, 16 total
Σ
#
Presentation
Properties
φ
4
14
⟨
a
,
b
|
abba
=1⟩
Grp
Inf
14
4
20
⟨
a
,
b
|
aba
=
b
⟩
Can
Inf