#5311 ⟨a, b | aaa=ab, aab=aa⟩

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 ⋅ a³ = a² and a ⋅ a = a², however a³ ≠ a
Not right cancellative, because right multiplication by a is not injective:
- a³ ⋅ a = a² and a ⋅ a = a², however a³ ≠ a

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.	a⁴ ⇒ a²	[2]
2.	ab ⇒ a³	[1]

# ab:aaa=ab,aab=aa a/b - -
aaaa=aa
ab=aaa

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

5 total

Σ	#	Presentation	Mapping
10	5315	⟨a, b \| aaa=ab, aba=aa⟩	φ(a) = a, φ(b) = b
11	15981	⟨a, b \| aaa=ab, aaaa=aa⟩	φ(a) = a, φ(b) = b
11	16086	⟨a, b \| aab=aa, aaab=ab⟩	φ(a) = a, φ(b) = b
11	16090	⟨a, b \| aab=aa, aaba=ab⟩	φ(a) = a, φ(b) = b
11	16345	⟨a, b \| aba=aa, aaba=ab⟩	φ(a) = a, φ(b) = b

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

1 total

Σ	#	Presentation	Mapping
11	16346	⟨a, b \| aba=aa, aaba=ba⟩	φ(a) = a, φ(b) = b