#2576 ⟨a, b | aaa=1, abab=ab⟩

Properties

Sum of relation sides is 9
Infinite non-cancellative non-commutative monoid
Element ba² has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ ab = b and b ⋅ 1 = b, however ab ≠ 1
Not right cancellative, because right multiplication by b is not injective:
- ba ⋅ b = b and 1 ⋅ b = b, however ba ≠ 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.	bab ⇒ b	[3]

# ab:aaa=1,abab=ab ab - -
aaa=1
bab=b

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

5 total

Σ	#	Presentation	Mapping
10	7491	⟨a, b \| aaa=1, aaabab=b⟩	φ(a) = a, φ(b) = b
10	7533	⟨a, b \| aaa=1, baaaab=b⟩	φ(a) = a, φ(b) = b
10	8035	⟨a, b \| aaa=1, aaab=bab⟩	φ(a) = a, φ(b) = b
11	22757	⟨a, b \| aaa=1, aabab=aab⟩	φ(a) = a, φ(b) = b
11	22790	⟨a, b \| aaa=1, ababa=aba⟩	φ(a) = a, φ(b) = b