#12413 ⟨
a
,
b
|
aaba
=
bb
,
abbb
=
b
⟩
Up:
Monoids with two generators and two relations
Prev:
#12403
⟨
a
,
b
|
aaba
=
ba
,
bbbb
=
b
⟩
Next:
#12415
⟨
a
,
b
|
aaba
=
bb
,
baaa
=
b
⟩
Quick links
Properties
Staircase diagram
Rewriting system
Properties
Presentation has sum-of-sides 11
Infinite non-cancellative commutative monoid
Not cancellative, because multiplication by
b
6
is not injective:
b
6
⋅
b
3
=
b
2
and
b
6
⋅
a
2
=
b
2
, however
b
3
≠
a
2
Commutative Gröbner basis: ⟨
a
,
b
|
b
8
=
b
,
ab
=
b
6
⟩
Cancellative quotient is isomorphic to ℤ
7
Enveloping group is isomorphic to ℤ
7
Group of units is isomorphic to ℤ
1
3 Archimedian components:
1,
a
,
b
Staircase diagram
Rewriting system
Format:
Pretty
Plain
Word to reduce:
Tips:
Lowercase letters stand for generators.
Spaces are ignored.
Numbers repeat the previous letter, e.g.
b90
.
Reduction strategy:
Leftmost
Rightmost
Path to normal form:
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
8
⇒
b
[10]
2.
ab
⇒
b
6
[9]
3.
ba
⇒
b
6
[11]
# ab:aaba=bb,abbb=b b/a bbbbbbbb=b ab=bbbbbb ba=bbbbbb
