#2583 ⟨a, b | aaa=1, abbb=ab⟩

Properties

Sum of relation sides is 9
Isomorphic to ℤ₃ ∗ ℕ_{(3 = 1)}
Infinite non-cancellative non-commutative monoid
Element aba has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ b² = b and b ⋅ 1 = b, however b² ≠ 1
Not right cancellative, because right multiplication by b is not injective:
- b² ⋅ b = b and 1 ⋅ b = b, however b² ≠ 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.	b³ ⇒ b	[3]

# ab:aaa=1,abbb=ab ab - -
aaa=1
bbb=b

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

5 total

Σ	#	Presentation	Mapping
10	7495	⟨a, b \| aaa=1, aaabbb=b⟩	φ(a) = a, φ(b) = b
10	7535	⟨a, b \| aaa=1, baaabb=b⟩	φ(a) = a, φ(b) = b
10	8037	⟨a, b \| aaa=1, aaab=bbb⟩	φ(a) = a, φ(b) = b
11	22773	⟨a, b \| aaa=1, aabbb=aab⟩	φ(a) = a, φ(b) = b
11	22812	⟨a, b \| aaa=1, abbba=aba⟩	φ(a) = a, φ(b) = b