#20 ⟨
a
,
b
|
aba
=
b
⟩
Up:
Monoid enumeration
Prev:
#19
⟨
a
,
b
|
aba
=
a
⟩
Next:
#21
⟨
a
,
b
|
ab
=
aa
⟩
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
) =
ba
,
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.
aba
⇒
b
[1]
2.
b
2
a
⇒
a
b
2
[2]
# ab:aba=b ab - - aba=b bba=abb
Other submonoids of same group
2 unique, 16 total
Σ
#
Presentation
Properties
φ
4
14
⟨
a
,
b
|
abba
=1⟩
Grp
Inf
14
4
23
⟨
a
,
b
|
bb
=
aa
⟩
Can
Inf