#14632 ⟨
a
,
b
|
aaba
=
b
,
abbbb
=
b
⟩
Up:
Monoids with two generators and two relations
Prev:
#14631
⟨
a
,
b
|
aaba
=
b
,
abbbb
=
a
⟩
Next:
#14634
⟨
a
,
b
|
aaba
=
b
,
baaaa
=
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
ab
is not injective:
ab
⋅
b
3
=
b
and
ab
⋅
a
2
=
b
, however
b
3
≠
a
2
Commutative Gröbner basis: ⟨
a
,
b
|
b
4
=
a
2
b
,
a
3
b
=
b
⟩
Cancellative quotient is isomorphic to ℤ
9
Enveloping group is isomorphic to ℤ
9
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
10
⇒
b
[9]
2.
ab
⇒
b
7
[8]
3.
ba
⇒
b
7
[10]
# ab:aaba=b,abbbb=b b/a bbbbbbbbbb=b ab=bbbbbbb ba=bbbbbbb
