#15487 ⟨
a
,
b
|
aaa
=
ab
,
babbb
=
a
⟩
Up:
Monoids with two generators and two relations
Prev:
#15479
⟨
a
,
b
|
aaa
=
ab
,
baabb
=
a
⟩
Next:
#15488
⟨
a
,
b
|
aaa
=
ab
,
babbb
=
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
⋅
a
2
=
a
b
2
and
ab
⋅
b
=
a
b
2
, however
a
2
≠
b
Commutative Gröbner basis: ⟨
a
,
b
|
a
3
=
ab
,
a
b
4
=
a
⟩
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(
a
) = 0; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
9
⇒
a
[5]
2.
ba
⇒
a
3
[6]
3.
ab
⇒
a
3
[1]
# ab:aaa=ab,babbb=a a/b aaaaaaaaa=a ba=aaa ab=aaa
