#1984 ⟨a, b | aaa=a, aaba=a⟩

Properties

Sum of relation sides is 9
Infinite non-cancellative non-commutative monoid
Element a²b² 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.	a³ ⇒ a	[1]
2.	aba ⇒ a²	[3]

# ab:aaa=a,aaba=a ab - -
aaa=a
aba=aa

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

12 total

Σ	#	Presentation	Mapping
10	5226	⟨a, b \| aba=aa, aaba=a⟩	φ(a) = a, φ(b) = b
11	12150	⟨a, b \| aaaa=aa, aaba=a⟩	φ(a) = a, φ(b) = b
11	12242	⟨a, b \| aaab=ab, aaba=a⟩	φ(a) = a, φ(b) = b
11	12246	⟨a, b \| aaab=ab, abaa=a⟩	φ(a) = a, φ(b) = b
11	14557	⟨a, b \| aaba=a, ababa=a⟩	φ(a) = a, φ(b) = b
11	15819	⟨a, b \| aba=aa, ababa=a⟩	φ(a) = a, φ(b) = b
11	18692	⟨a, b \| aaa=a, aaaaba=a⟩	φ(a) = a, φ(b) = b
11	18696	⟨a, b \| aaa=a, aaabaa=a⟩	φ(a) = a, φ(b) = b
11	19239	⟨a, b \| aaa=a, aaaba=aa⟩	φ(a) = a, φ(b) = b
11	19247	⟨a, b \| aaa=a, aabaa=aa⟩	φ(a) = a, φ(b) = b
11	19778	⟨a, b \| aaa=a, aaaa=aba⟩	φ(a) = a, φ(b) = b
11	19790	⟨a, b \| aaa=a, aaba=aaa⟩	φ(a) = a, φ(b) = b