#1135 ⟨a, b | aa=1, bbbbb=b⟩

Properties

Sum of relation sides is 8
Isomorphic to ℤ₂ ∗ ℕ_{(5 = 1)}
Infinite non-cancellative non-commutative monoid
Element ba has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ b⁴ = b and b ⋅ 1 = b, however b⁴ ≠ 1
Not right cancellative, because right multiplication by b is not injective:
- b⁴ ⋅ b = b and 1 ⋅ b = b, however b⁴ ≠ 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.	b⁵ ⇒ b	[2]

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

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

5 total

Σ	#	Presentation	Mapping
10	9715	⟨a, b \| aa=1, aabbbbb=b⟩	φ(a) = a, φ(b) = b
10	9775	⟨a, b \| aa=1, baabbbb=b⟩	φ(a) = a, φ(b) = b
10	9791	⟨a, b \| aa=1, bbaabbb=b⟩	φ(a) = a, φ(b) = b
10	10025	⟨a, b \| aa=1, abbbbb=ab⟩	φ(a) = a, φ(b) = b
10	10339	⟨a, b \| aa=1, bbbbb=aab⟩	φ(a) = a, φ(b) = b