#3141 ⟨a, b | aa=a, abb=aba⟩

Properties

Sum of relation sides is 9
Infinite non-cancellative non-commutative monoid
Element ba 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.	ab² ⇒ aba	[2]

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

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

7 total

Σ	#	Presentation	Mapping
10	9129	⟨a, b \| aa=a, aaba=abb⟩	φ(a) = a, φ(b) = b
10	9136	⟨a, b \| aa=a, aabb=aba⟩	φ(a) = a, φ(b) = b
11	25133	⟨a, b \| aa=a, aaaba=abb⟩	φ(a) = a, φ(b) = b
11	25140	⟨a, b \| aa=a, aaabb=aba⟩	φ(a) = a, φ(b) = b
11	25149	⟨a, b \| aa=a, aabaa=abb⟩	φ(a) = a, φ(b) = b
11	25665	⟨a, b \| aa=a, aabb=aaba⟩	φ(a) = a, φ(b) = b
11	25668	⟨a, b \| aa=a, abaa=aabb⟩	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
10	9132	⟨a, b \| aa=a, aaba=bba⟩	φ(a) = a, φ(b) = b
11	25136	⟨a, b \| aa=a, aaaba=bba⟩	φ(a) = a, φ(b) = b