#892 ⟨a, b | aa=a, abbb=a⟩

Properties

Sum of relation sides is 8
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.	ab³ ⇒ a	[2]

# ab:aa=a,abbb=a ab - -
aa=a
abbb=a

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

12 total

Σ	#	Presentation	Mapping
9	2870	⟨a, b \| aa=a, aabbb=a⟩	φ(a) = a, φ(b) = b
9	3022	⟨a, b \| aa=a, abbb=aa⟩	φ(a) = a, φ(b) = b
10	5140	⟨a, b \| aab=ab, abbb=a⟩	φ(a) = a, φ(b) = b
10	8582	⟨a, b \| aa=a, aaabbb=a⟩	φ(a) = a, φ(b) = b
10	8866	⟨a, b \| aa=a, aabbb=aa⟩	φ(a) = a, φ(b) = b
10	9156	⟨a, b \| aa=a, abbb=aaa⟩	φ(a) = a, φ(b) = b
11	15415	⟨a, b \| aaa=aa, aabbb=a⟩	φ(a) = a, φ(b) = b
11	15623	⟨a, b \| aab=ab, aabbb=a⟩	φ(a) = a, φ(b) = b
11	24058	⟨a, b \| aa=a, aaaabbb=a⟩	φ(a) = a, φ(b) = b
11	24615	⟨a, b \| aa=a, aaabbb=aa⟩	φ(a) = a, φ(b) = b
11	25168	⟨a, b \| aa=a, aabbb=aaa⟩	φ(a) = a, φ(b) = b
11	25679	⟨a, b \| aa=a, abbb=aaaa⟩	φ(a) = a, φ(b) = b

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

3 total

Σ	#	Presentation	Mapping
11	14480	⟨a, b \| aaab=b, aaabb=b⟩	φ(a) = b, φ(b) = a
11	14508	⟨a, b \| aaab=b, baaab=b⟩	φ(a) = b, φ(b) = a
11	15665	⟨a, b \| aab=ab, bbbaa=a⟩	φ(a) = a, φ(b) = b