#8057 ⟨a, b | aaa=1, abab=abb⟩

Properties

Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element ba² has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ ab = b² and b ⋅ b = b², however ab ≠ b
Not right cancellative, because right multiplication by b is not injective:
- ba ⋅ b = b² and b ⋅ b = b², however ba ≠ b

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 ⇒ b²	[3]

# ab:aaa=1,abab=abb ab - -
aaa=1
bab=bb

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

4 total

Σ	#	Presentation	Mapping
11	22214	⟨a, b \| aaa=1, aaabab=bb⟩	φ(a) = a, φ(b) = b
11	22294	⟨a, b \| aaa=1, baaaab=bb⟩	φ(a) = a, φ(b) = b
11	22747	⟨a, b \| aaa=1, aaabb=bab⟩	φ(a) = a, φ(b) = b
11	22828	⟨a, b \| aaa=1, baaab=bab⟩	φ(a) = a, φ(b) = b