#5064 ⟨a, b | aaa=ab, babb=a⟩

Properties

Presentation has sum-of-sides 10
Infinite non-cancellative commutative monoid
Commutative Gröbner basis: ⟨a, b | a³=ab, ab³=a⟩
Not cancellative, because multiplication by ab is not injective:
- a² ≠ b
- ab ⋅ a² = ab²
- ab ⋅ b = ab²
- Cancellative quotient is isomorphic to ℤ₆
Grothendieck group is isomorphic to ℤ₆
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:aaa=ab,babb=a a/b
aaaaaaa=a
ba=aaa
ab=aaa

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

6 total

Σ	#	Presentation	Mapping
11	14624	⟨a, b \| aaba=b, ababb=b⟩	φ(a) = b, φ(b) = a
11	14628	⟨a, b \| aaba=b, abbab=b⟩	φ(a) = b, φ(b) = a
11	14630	⟨a, b \| aaba=b, abbba=b⟩	φ(a) = b, φ(b) = a
11	15475	⟨a, b \| aaa=ab, baaab=a⟩	φ(a) = a, φ(b) = b
11	15477	⟨a, b \| aaa=ab, baaba=a⟩	φ(a) = a, φ(b) = b
11	15481	⟨a, b \| aaa=ab, babaa=a⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
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Next:	#5065 ⟨a, b \| aaa=ab, babb=b⟩