#20862 ⟨a, b | ba=ab, aaabb=bb⟩

Properties

Presentation has sum-of-sides 11
Infinite non-cancellative commutative monoid
Commutative Gröbner basis: ⟨a, b | a³b²=b²⟩
Not cancellative, because multiplication by a²b² is not injective:
- a³ ≠ 1
- a²b² ⋅ a³ = a²b²
- a²b² ⋅ 1 = a²b²
- Cancellative quotient is isomorphic to ℤ₃ ⊕ ℕ
Grothendieck group is isomorphic to ℤ₃ ⊕ 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,aaabb=bb ab
ba=ab
aaabb=bb

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

5 total

Σ	#	Presentation	Mapping
11	20869	⟨a, b \| ba=ab, aabab=bb⟩	φ(a) = a, φ(b) = b
11	20873	⟨a, b \| ba=ab, aabba=bb⟩	φ(a) = a, φ(b) = b
11	20877	⟨a, b \| ba=ab, abaab=bb⟩	φ(a) = a, φ(b) = b
11	20880	⟨a, b \| ba=ab, ababa=bb⟩	φ(a) = a, φ(b) = b
11	20881	⟨a, b \| ba=ab, abbba=aa⟩	φ(a) = b, φ(b) = a

Up:	Monoids with two generators and two relations
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Next:	#20884 ⟨a, b \| bb=aa, aaaaa=aa⟩