#753 ⟨a, b | aaa=1, babb=b⟩

Properties

Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element ba² has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ ab² = b and b ⋅ 1 = b, however ab² ≠ 1
Not right cancellative, because right multiplication by b is not injective:
- bab ⋅ b = b and 1 ⋅ b = b, however bab ≠ 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 ⇒ 1	[1]
2.	bab² ⇒ b	[2]

# ab:aaa=1,babb=b ab - -
aaa=1
babb=b

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

5 total

Σ	#	Presentation	Mapping
10	7790	⟨a, b \| aaa=1, ababb=ab⟩	φ(a) = a, φ(b) = b
11	21671	⟨a, b \| aaa=1, aaababb=b⟩	φ(a) = a, φ(b) = b
11	21755	⟨a, b \| aaa=1, baaaabb=b⟩	φ(a) = a, φ(b) = b
11	23305	⟨a, b \| aaa=1, babb=aaab⟩	φ(a) = a, φ(b) = b
11	23312	⟨a, b \| aaa=1, babb=baaa⟩	φ(a) = a, φ(b) = b

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

3 total

Σ	#	Presentation	Mapping
10	7794	⟨a, b \| aaa=1, abbab=ab⟩	φ(a) = a, φ(b) = b
11	21675	⟨a, b \| aaa=1, aaabbab=b⟩	φ(a) = a, φ(b) = b
11	21757	⟨a, b \| aaa=1, baaabab=b⟩	φ(a) = a, φ(b) = b