#4134 ⟨a, b | aba=aaa, baa=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 ⋅ 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.	ba ⇒ a²	[5]
2.	a³ ⇒ a	[4]

# ab:aba=aaa,baa=a ab - -
ba=aa
aaa=a

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

8 total

Σ	#	Presentation	Mapping
10	6389	⟨a, b \| aab=b, aaabb=b⟩	φ(a) = b, φ(b) = a
10	6401	⟨a, b \| aab=b, abaab=b⟩	φ(a) = b, φ(b) = a
10	6669	⟨a, b \| aab=b, aaab=bb⟩	φ(a) = b, φ(b) = a
10	6675	⟨a, b \| aab=b, aabb=ab⟩	φ(a) = b, φ(b) = a
10	6699	⟨a, b \| aab=b, baab=ab⟩	φ(a) = b, φ(b) = a
10	6735	⟨a, b \| aba=a, aaba=ba⟩	φ(a) = a, φ(b) = b
10	6819	⟨a, b \| aab=b, abb=aab⟩	φ(a) = b, φ(b) = a
10	6843	⟨a, b \| abb=b, bbb=abb⟩	φ(a) = b, φ(b) = a