#5222 ⟨a, b | aba=aa, aaaa=a⟩

Properties

Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element a³b² has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ a³ = a and a ⋅ 1 = a, however a³ ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- a³ ⋅ a = a and 1 ⋅ a = a, however a³ ≠ 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.	aba ⇒ a²	[1]
2.	a⁴ ⇒ a	[2]

# ab:aba=aa,aaaa=a ab - -
aba=aa
aaaa=a

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

4 total

Σ	#	Presentation	Mapping
11	14333	⟨a, b \| aaaa=a, aaaba=a⟩	φ(a) = a, φ(b) = b
11	14337	⟨a, b \| aaaa=a, aabaa=a⟩	φ(a) = a, φ(b) = b
11	15805	⟨a, b \| aba=aa, aaaba=a⟩	φ(a) = a, φ(b) = b
11	15809	⟨a, b \| aba=aa, aabaa=a⟩	φ(a) = a, φ(b) = b