#565 ⟨a, b | aaa=a, aba=a⟩

Properties

Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element a²b² has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ a² = a and a ⋅ 1 = a, however a² ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- a² ⋅ a = a and 1 ⋅ a = a, however a² ≠ 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.	a³ ⇒ a	[1]
2.	aba ⇒ a	[2]

# ab:aaa=a,aba=a ab - -
aaa=a
aba=a

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

7 total

Σ	#	Presentation	Mapping
10	6242	⟨a, b \| aaa=a, aaaba=a⟩	φ(a) = a, φ(b) = b
10	6246	⟨a, b \| aaa=a, aabaa=a⟩	φ(a) = a, φ(b) = b
10	6450	⟨a, b \| aba=a, aaaba=a⟩	φ(a) = a, φ(b) = b
10	6454	⟨a, b \| aba=a, aabaa=a⟩	φ(a) = a, φ(b) = b
10	6533	⟨a, b \| aaa=a, aaba=aa⟩	φ(a) = a, φ(b) = b
10	6730	⟨a, b \| aba=a, aaab=ab⟩	φ(a) = a, φ(b) = b
10	6799	⟨a, b \| aaa=a, aba=aaa⟩	φ(a) = a, φ(b) = b