#5048 ⟨a, b | aaa=ab, aabb=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 ⋅ a⁵ = a and a ⋅ 1 = a, however a⁵ ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- a⁵ ⋅ a = a and 1 ⋅ a = a, however a⁵ ≠ 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.	a⁶ ⇒ a	[2]
2.	ab ⇒ a³	[1]

# ab:aaa=ab,aabb=a a/b - -
aaaaaa=a
ab=aaa

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

12 total

Σ	#	Presentation	Mapping
10	5052	⟨a, b \| aaa=ab, abab=a⟩	φ(a) = a, φ(b) = b
10	5054	⟨a, b \| aaa=ab, abba=a⟩	φ(a) = a, φ(b) = b
11	14667	⟨a, b \| aabb=a, aaaab=a⟩	φ(a) = a, φ(b) = b
11	14731	⟨a, b \| abab=a, aaaab=a⟩	φ(a) = a, φ(b) = b
11	14733	⟨a, b \| abab=a, aaaba=a⟩	φ(a) = a, φ(b) = b
11	14737	⟨a, b \| abab=a, aabaa=a⟩	φ(a) = a, φ(b) = b
11	14745	⟨a, b \| abab=a, abaaa=a⟩	φ(a) = a, φ(b) = b
11	14795	⟨a, b \| abba=a, aaaab=a⟩	φ(a) = a, φ(b) = b
11	15443	⟨a, b \| aaa=ab, aaaab=a⟩	φ(a) = a, φ(b) = b
11	15445	⟨a, b \| aaa=ab, aaaba=a⟩	φ(a) = a, φ(b) = b
11	15449	⟨a, b \| aaa=ab, aabaa=a⟩	φ(a) = a, φ(b) = b
11	15457	⟨a, b \| aaa=ab, abaaa=a⟩	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
11	12301	⟨a, b \| aaab=bb, aabb=b⟩	φ(a) = b, φ(b) = a
11	12305	⟨a, b \| aaab=bb, abab=b⟩	φ(a) = b, φ(b) = a