#7129 ⟨a, b | ba=ab, aabb=aa⟩

Properties

Presentation has sum-of-sides 10
Infinite non-cancellative commutative monoid
Commutative Gröbner basis: ⟨a, b | a²b²=a²⟩
Not cancellative, because multiplication by a² is not injective:
- b² ≠ 1
- a² ⋅ b² = a²
- a² ⋅ 1 = a²
- Cancellative quotient is isomorphic to ℤ₂ ⊕ ℕ
Grothendieck group is isomorphic to ℤ₂ ⊕ ℤ
Group of units is isomorphic to ℤ₁
3 Archimedian components:
- 1, a, b

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

# ab:ba=ab,aabb=aa ab
ba=ab
aabb=aa

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

3 total

Σ	#	Presentation	Mapping
10	7132	⟨a, b \| ba=ab, abab=aa⟩	φ(a) = a, φ(b) = b
10	7135	⟨a, b \| ba=ab, abba=aa⟩	φ(a) = a, φ(b) = b
10	7137	⟨a, b \| ba=ab, abba=bb⟩	φ(a) = b, φ(b) = a

Up:	Monoids with two generators and two relations
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Next:	#7130 ⟨a, b \| ba=ab, aabb=ab⟩