#5312 ⟨a, b | aaa=ab, aab=ab⟩

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:
- a³ ⋅ a = a³ and a² ⋅ a = a³, however a³ ≠ a²

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.	a⁴ ⇒ a³	[2]
2.	ab ⇒ a³	[1]

# ab:aaa=ab,aab=ab a/b - -
aaaa=aaa
ab=aaa

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

4 total

Σ	#	Presentation	Mapping
10	5316	⟨a, b \| aaa=ab, aba=ab⟩	φ(a) = a, φ(b) = b
11	15982	⟨a, b \| aaa=ab, aaaa=ab⟩	φ(a) = a, φ(b) = b
11	16478	⟨a, b \| aaa=ab, aab=aaa⟩	φ(a) = a, φ(b) = b
11	16479	⟨a, b \| aaa=ab, aba=aaa⟩	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	16554	⟨a, b \| abb=ab, bbb=abb⟩	φ(a) = b, φ(b) = a
11	16558	⟨a, b \| bab=ab, bbb=bab⟩	φ(a) = b, φ(b) = a