#14872 ⟨
a
,
b
|
abba
=
b
,
bbbbb
=
b
⟩
Up:
Monoids with two generators and two relations
Prev:
#14869
⟨
a
,
b
|
abba
=
b
,
bbabb
=
a
⟩
Next:
#14908
⟨
a
,
b
|
aaa
=
aa
,
bbbbbb
=1⟩
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
=
a
b
4
and
ab
⋅
a
2
=
a
b
4
, however
b
3
≠
a
2
Commutative Gröbner basis: ⟨
a
,
b
|
b
5
=
b
,
a
2
b
=
b
4
⟩
Cancellative quotient is isomorphic to ℤ
8
Enveloping group is isomorphic to ℤ
8
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
5
⇒
b
[2]
2.
ba
⇒
ab
[5]
3.
a
2
b
⇒
b
4
[7]
# ab:abba=b,bbbbb=b b/a bbbbb=b ba=ab aab=bbbb
