#22 ⟨
a
,
b
|
ba
=
ab
⟩
Up:
Monoids with two generators and one relation
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#21
⟨
a
,
b
|
ab
=
aa
⟩
Next:
#23
⟨
a
,
b
|
bb
=
aa
⟩
Quick links
Properties
Staircase diagram
Rewriting system
Properties
Presentation has sum-of-sides 4
Isomorphic to ℕ ⊕ ℕ
Infinite cancellative commutative monoid
Commutative Gröbner basis: ⟨
a
,
b
| ⟩
Enveloping group is isomorphic to ℤ ⊕ 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]
# ab:ba=ab ab ba=ab
