#20248 ⟨
a
,
b
|
aba
=
b
,
aaaa
=
aaa
⟩
Up:
Monoids with two generators and two relations
Prev:
#20247
⟨
a
,
b
|
aba
=
a
,
bbbb
=
bbb
⟩
Next:
#20251
⟨
a
,
b
|
aba
=
b
,
aaaa
=
abb
⟩
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
is not injective:
b
⋅
a
=
b
and
b
⋅ 1 =
b
, however
a
≠ 1
Commutative Gröbner basis: ⟨
a
,
b
|
ab
=
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
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 shortlex with
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ab
⇒
b
[5]
2.
ba
⇒
b
[6]
3.
a
4
⇒
a
3
[2]
# ab:aba=b,aaaa=aaa ab ab=b ba=b aaaa=aaa
