#2874 ⟨a, b | aa=a, ababa=a⟩

Properties

Sum of relation sides is 9
Infinite non-cancellative non-commutative monoid
Element abab² has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ a = a and a ⋅ 1 = a, however a ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- a ⋅ a = a and 1 ⋅ a = a, however a ≠ 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.	a² ⇒ a	[1]
2.	a(ba)² ⇒ a	[2]

# ab:aa=a,ababa=a ab - -
aa=a
ababa=a

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

11 total

Σ	#	Presentation	Mapping
10	8586	⟨a, b \| aa=a, aababa=a⟩	φ(a) = a, φ(b) = b
10	8600	⟨a, b \| aa=a, abaaba=a⟩	φ(a) = a, φ(b) = b
10	8874	⟨a, b \| aa=a, ababa=aa⟩	φ(a) = a, φ(b) = b
11	15629	⟨a, b \| aab=ab, ababa=a⟩	φ(a) = a, φ(b) = b
11	24064	⟨a, b \| aa=a, aaababa=a⟩	φ(a) = a, φ(b) = b
11	24078	⟨a, b \| aa=a, aabaaba=a⟩	φ(a) = a, φ(b) = b
11	24082	⟨a, b \| aa=a, aababaa=a⟩	φ(a) = a, φ(b) = b
11	24106	⟨a, b \| aa=a, abaaaba=a⟩	φ(a) = a, φ(b) = b
11	24623	⟨a, b \| aa=a, aababa=aa⟩	φ(a) = a, φ(b) = b
11	24650	⟨a, b \| aa=a, abaaba=aa⟩	φ(a) = a, φ(b) = b
11	25184	⟨a, b \| aa=a, ababa=aaa⟩	φ(a) = a, φ(b) = b