#663 ⟨
a
,
b
|
ba
=
ab
,
bb
=
aa
⟩
Up:
Monoids with two generators and two relations
Prev:
#662
⟨
a
,
b
|
ab
=
aa
,
bb
=
ba
⟩
Next:
#683
⟨
a
,
b
|
aaa
=1,
bbbbb
=1⟩
Properties
Presentation has sum-of-sides 8
Isomorphic to ⟨(1, 1), (0, 1)⟩ ⊂ ℤ
2
⊕ ℤ
Infinite cancellative commutative monoid
Commutative Gröbner basis: ⟨
a
,
b
|
a
2
=
b
2
⟩
Grothendieck group is isomorphic to ℤ
2
⊕ ℤ
Group of units is isomorphic to ℤ
1
2 Archimedian components:
1,
b
Complete 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
ba
⇒
ab
b
2
⇒
a
2
# ab:ba=ab,bb=aa ab ba=ab bb=aa
Staircase diagram
Right Cayley graph (truncated)
