#3407 ⟨a, b | aa=1, bbbbbb=b⟩

Properties

Sum of relation sides is 9
Isomorphic to ℤ₂ ∗ ℕ_{(6 = 1)}
Infinite non-cancellative non-commutative monoid
Element ba has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ b⁵ = b and b ⋅ 1 = b, however b⁵ ≠ 1
Not right cancellative, because right multiplication by b is not injective:
- b⁵ ⋅ b = b and 1 ⋅ b = b, however b⁵ ≠ 1

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

#	Rule	Proof
1.	a² ⇒ 1	[1]
2.	b⁶ ⇒ b	[2]

# ab:aa=1,bbbbbb=b ab - -
aa=1
bbbbbb=b

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

6 total

Σ	#	Presentation	Mapping
11	26555	⟨a, b \| aa=1, aabbbbbb=b⟩	φ(a) = a, φ(b) = b
11	26669	⟨a, b \| aa=1, baabbbbb=b⟩	φ(a) = a, φ(b) = b
11	26699	⟨a, b \| aa=1, bbaabbbb=b⟩	φ(a) = a, φ(b) = b
11	26707	⟨a, b \| aa=1, bbbaabbb=b⟩	φ(a) = a, φ(b) = b
11	27165	⟨a, b \| aa=1, abbbbbb=ab⟩	φ(a) = a, φ(b) = b
11	27763	⟨a, b \| aa=1, bbbbbb=aab⟩	φ(a) = a, φ(b) = b