#5056 ⟨a, b | aaa=ab, abbb=a⟩

Properties

Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element b has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ b = a³ and a ⋅ a² = a³, however b ≠ a²
Not right cancellative, because right multiplication by a is not injective:
- a⁶ ⋅ a = a and 1 ⋅ a = a, however a⁶ ≠ 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.	a⁷ ⇒ a	[2]
2.	ab ⇒ a³	[1]

# ab:aaa=ab,abbb=a a/b - -
aaaaaaa=a
ab=aaa

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

6 total

Σ	#	Presentation	Mapping
11	15447	⟨a, b \| aaa=ab, aaabb=a⟩	φ(a) = a, φ(b) = b
11	15451	⟨a, b \| aaa=ab, aabab=a⟩	φ(a) = a, φ(b) = b
11	15453	⟨a, b \| aaa=ab, aabba=a⟩	φ(a) = a, φ(b) = b
11	15459	⟨a, b \| aaa=ab, abaab=a⟩	φ(a) = a, φ(b) = b
11	15461	⟨a, b \| aaa=ab, ababa=a⟩	φ(a) = a, φ(b) = b
11	15465	⟨a, b \| aaa=ab, abbaa=a⟩	φ(a) = a, φ(b) = b

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

2 total

Σ	#	Presentation	Mapping
11	14488	⟨a, b \| aaab=b, aabbb=b⟩	φ(a) = b, φ(b) = a
11	14500	⟨a, b \| aaab=b, abbab=b⟩	φ(a) = b, φ(b) = a