#21048 ⟨
a
,
b
|
ba
=
ab
,
aaaa
=
aaa
⟩
Up:
Monoids with two generators and two relations
Prev:
#21047
⟨
a
,
b
|
ab
=
aa
,
bbbb
=
bbb
⟩
Next:
#21049
⟨
a
,
b
|
ba
=
ab
,
aaaa
=
aab
⟩
Quick links
Properties
Staircase diagram
Rewriting system
Properties
Presentation has sum-of-sides 11
Isomorphic to ℕ
(4 = 3)
⊕ ℕ
Infinite non-cancellative commutative monoid
Not cancellative, because multiplication by
a
3
b
3
is not injective:
a
3
b
3
⋅
a
=
a
3
b
3
and
a
3
b
3
⋅ 1 =
a
3
b
3
, however
a
≠ 1
Commutative Gröbner basis: ⟨
a
,
b
|
a
4
=
a
3
⟩
Cancellative quotient is isomorphic to ℕ
Enveloping group is isomorphic to ℤ
Group of units is isomorphic to ℤ
1
4 Archimedian components:
1,
a
,
b
,
ab
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 shortlex with
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ba
⇒
ab
[1]
2.
a
4
⇒
a
3
[2]
# ab:ba=ab,aaaa=aaa ab ba=ab aaaa=aaa
