#7196 ⟨
a
,
b
|
ba
=
ab
,
bbb
=
aaa
⟩
Up:
Monoids with two generators and two relations
Prev:
#7190
⟨
a
,
b
|
ba
=
ab
,
abb
=
aab
⟩
Next:
#7197
⟨
a
,
b
|
bb
=
aa
,
aab
=
aaa
⟩
Quick links
Properties
Staircase diagram
Rewriting system
Properties
Presentation has sum-of-sides 10
Isomorphic to ⟨(1, 1), (0, 1)⟩ ⊂ ℤ
3
⊕ ℤ
Infinite cancellative commutative monoid
Commutative Gröbner basis: ⟨
a
,
b
|
a
3
=
b
3
⟩
Enveloping group is isomorphic to ℤ
3
⊕ ℤ
Group of units is isomorphic to ℤ
1
2 Archimedian components:
1,
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.
ba
⇒
ab
[1]
2.
b
3
⇒
a
3
[2]
# ab:ba=ab,bbb=aaa ab ba=ab bbb=aaa
