#5319 ⟨a, b | aaa=ab, abb=aa⟩

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:
- a⁴ ⋅ a = a² and a ⋅ a = a², however a⁴ ≠ a

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.	a⁵ ⇒ a²	[2]
2.	ab ⇒ a³	[1]

# ab:aaa=ab,abb=aa a/b - -
aaaaa=aa
ab=aaa

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

3 total

Σ	#	Presentation	Mapping
11	15985	⟨a, b \| aaa=ab, aaab=aa⟩	φ(a) = a, φ(b) = b
11	15989	⟨a, b \| aaa=ab, aaba=aa⟩	φ(a) = a, φ(b) = b
11	15997	⟨a, b \| aaa=ab, abaa=aa⟩	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	16286	⟨a, b \| aab=bb, aabb=ab⟩	φ(a) = b, φ(b) = a
11	16310	⟨a, b \| aab=bb, baab=ab⟩	φ(a) = b, φ(b) = a