#755 ⟨a, b | aaa=1, bbbb=b⟩

Properties

Sum of relation sides is 8
Isomorphic to ℤ₃ ∗ ℕ_{(4 = 1)}
Infinite non-cancellative non-commutative monoid
Element aba has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ b³ = b and b ⋅ 1 = b, however b³ ≠ 1
Not right cancellative, because right multiplication by b is not injective:
- b³ ⋅ b = b and 1 ⋅ b = b, however b³ ≠ 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.	a³ ⇒ 1	[1]
2.	b⁴ ⇒ b	[2]

# ab:aaa=1,bbbb=b ab - -
aaa=1
bbbb=b

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

5 total

Σ	#	Presentation	Mapping
10	7801	⟨a, b \| aaa=1, abbbb=ab⟩	φ(a) = a, φ(b) = b
11	21679	⟨a, b \| aaa=1, aaabbbb=b⟩	φ(a) = a, φ(b) = b
11	21759	⟨a, b \| aaa=1, baaabbb=b⟩	φ(a) = a, φ(b) = b
11	21781	⟨a, b \| aaa=1, bbaaabb=b⟩	φ(a) = a, φ(b) = b
11	23322	⟨a, b \| aaa=1, bbbb=aaab⟩	φ(a) = a, φ(b) = b