#1117 ⟨a, b | aa=1, ababb=b⟩

Properties

Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element b 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

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

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

11 total

Σ	#	Presentation	Mapping
8	1198	⟨a, b \| aa=1, babb=ab⟩	φ(a) = a, φ(b) = b
10	9679	⟨a, b \| aa=1, aaababb=b⟩	φ(a) = a, φ(b) = b
10	9721	⟨a, b \| aa=1, abaaabb=b⟩	φ(a) = a, φ(b) = b
10	9729	⟨a, b \| aa=1, ababaab=b⟩	φ(a) = a, φ(b) = b
10	9968	⟨a, b \| aa=1, aababb=ab⟩	φ(a) = a, φ(b) = b
10	10003	⟨a, b \| aa=1, ababba=ba⟩	φ(a) = a, φ(b) = b
10	10032	⟨a, b \| aa=1, baaabb=ab⟩	φ(a) = a, φ(b) = b
10	10275	⟨a, b \| aa=1, ababb=aab⟩	φ(a) = a, φ(b) = b
10	10278	⟨a, b \| aa=1, ababb=baa⟩	φ(a) = a, φ(b) = b
10	10513	⟨a, b \| aa=1, babb=aaab⟩	φ(a) = a, φ(b) = b
10	10516	⟨a, b \| aa=1, babb=abaa⟩	φ(a) = a, φ(b) = b

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

4 total

Σ	#	Presentation	Mapping
10	9705	⟨a, b \| aa=1, aabbaba=b⟩	φ(a) = a, φ(b) = b
10	9976	⟨a, b \| aa=1, aabbab=ba⟩	φ(a) = a, φ(b) = b
10	10037	⟨a, b \| aa=1, baabab=ba⟩	φ(a) = a, φ(b) = b
10	10284	⟨a, b \| aa=1, abbab=aba⟩	φ(a) = a, φ(b) = b