#1129 ⟨a, b | aa=1, babab=b⟩

Properties

Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element ab² has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ (ab)² = b and b ⋅ 1 = b, however (ab)² ≠ 1
Not right cancellative, because right multiplication by b is not injective:
- (ba)² ⋅ b = b and 1 ⋅ b = b, however (ba)² ≠ 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.	a² ⇒ 1	[1]
2.	b(ab)² ⇒ b	[2]

# ab:aa=1,babab=b ab - -
aa=1
babab=b

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

4 total

Σ	#	Presentation	Mapping
10	9697	⟨a, b \| aa=1, aababab=b⟩	φ(a) = a, φ(b) = b
10	9765	⟨a, b \| aa=1, baaabab=b⟩	φ(a) = a, φ(b) = b
10	9998	⟨a, b \| aa=1, ababab=ab⟩	φ(a) = a, φ(b) = b
10	10319	⟨a, b \| aa=1, babab=aab⟩	φ(a) = a, φ(b) = b