#40 ⟨a, b | aa=a, bbb=1⟩

Properties

Sum of relation sides is 6
Isomorphic to ℕ_{(2 = 1)} ∗ ℤ₃
Infinite non-cancellative non-commutative monoid
Element ba has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ a = a and a ⋅ 1 = a, however a ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- a ⋅ a = a and 1 ⋅ a = a, however a ≠ 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 ⇒ a	[1]
2.	b³ ⇒ 1	[2]

# ab:aa=a,bbb=1 ab - -
aa=a
bbb=1

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

11 total

Σ	#	Presentation	Mapping
8	419	⟨a, b \| aab=ab, bbb=1⟩	φ(a) = a, φ(b) = b
9	2423	⟨a, b \| aaa=1, aaabb=b⟩	φ(a) = b, φ(b) = a
9	2445	⟨a, b \| aaa=1, baaab=b⟩	φ(a) = b, φ(b) = a
9	2566	⟨a, b \| aaa=1, aaab=bb⟩	φ(a) = b, φ(b) = a
10	8047	⟨a, b \| aaa=1, aabb=aab⟩	φ(a) = b, φ(b) = a
10	8064	⟨a, b \| aaa=1, abba=aba⟩	φ(a) = b, φ(b) = a
11	22204	⟨a, b \| aaa=1, aaaabb=ab⟩	φ(a) = b, φ(b) = a
11	22217	⟨a, b \| aaa=1, aaabba=ba⟩	φ(a) = b, φ(b) = a
11	22251	⟨a, b \| aaa=1, abaaab=ab⟩	φ(a) = b, φ(b) = a
11	22729	⟨a, b \| aaa=1, aaaab=abb⟩	φ(a) = b, φ(b) = a
11	22740	⟨a, b \| aaa=1, aaaba=bba⟩	φ(a) = b, φ(b) = a