#336 ⟨a, b | aa=1, babb=b⟩

Properties

Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element ba 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.	a² ⇒ 1	[1]
2.	bab² ⇒ b	[2]

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

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

24 total

Σ	#	Presentation	Mapping
9	3357	⟨a, b \| aa=1, aababb=b⟩	φ(a) = a, φ(b) = b
9	3391	⟨a, b \| aa=1, baaabb=b⟩	φ(a) = a, φ(b) = b
9	3510	⟨a, b \| aa=1, ababb=ab⟩	φ(a) = a, φ(b) = b
9	3667	⟨a, b \| aa=1, babb=aab⟩	φ(a) = a, φ(b) = b
9	3670	⟨a, b \| aa=1, babb=baa⟩	φ(a) = a, φ(b) = b
11	26463	⟨a, b \| aa=1, aaaababb=b⟩	φ(a) = a, φ(b) = b
11	26507	⟨a, b \| aa=1, aabaaabb=b⟩	φ(a) = a, φ(b) = b
11	26517	⟨a, b \| aa=1, aababaab=b⟩	φ(a) = a, φ(b) = b
11	26523	⟨a, b \| aa=1, aababbaa=b⟩	φ(a) = a, φ(b) = b
11	26643	⟨a, b \| aa=1, baaaaabb=b⟩	φ(a) = a, φ(b) = b
11	26649	⟨a, b \| aa=1, baaabaab=b⟩	φ(a) = a, φ(b) = b
11	27011	⟨a, b \| aa=1, aaababb=ab⟩	φ(a) = a, φ(b) = b
11	27051	⟨a, b \| aa=1, aababba=ba⟩	φ(a) = a, φ(b) = b
11	27092	⟨a, b \| aa=1, abaaabb=ab⟩	φ(a) = a, φ(b) = b
11	27108	⟨a, b \| aa=1, ababaab=ab⟩	φ(a) = a, φ(b) = b
11	27575	⟨a, b \| aa=1, aababb=aab⟩	φ(a) = a, φ(b) = b
11	27578	⟨a, b \| aa=1, aababb=baa⟩	φ(a) = a, φ(b) = b
11	27644	⟨a, b \| aa=1, ababba=aba⟩	φ(a) = a, φ(b) = b
11	27703	⟨a, b \| aa=1, baaabb=aab⟩	φ(a) = a, φ(b) = b
11	27706	⟨a, b \| aa=1, baaabb=baa⟩	φ(a) = a, φ(b) = b
11	28045	⟨a, b \| aa=1, aaaab=babb⟩	φ(a) = a, φ(b) = b
11	28090	⟨a, b \| aa=1, aabaa=babb⟩	φ(a) = a, φ(b) = b
11	28167	⟨a, b \| aa=1, ababb=aaab⟩	φ(a) = a, φ(b) = b
11	28170	⟨a, b \| aa=1, ababb=abaa⟩	φ(a) = a, φ(b) = b

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

18 total

Σ	#	Presentation	Mapping
9	3361	⟨a, b \| aa=1, aabbab=b⟩	φ(a) = a, φ(b) = b
9	3393	⟨a, b \| aa=1, baabab=b⟩	φ(a) = a, φ(b) = b
9	3514	⟨a, b \| aa=1, abbab=ab⟩	φ(a) = a, φ(b) = b
11	26467	⟨a, b \| aa=1, aaaabbab=b⟩	φ(a) = a, φ(b) = b
11	26511	⟨a, b \| aa=1, aabaabab=b⟩	φ(a) = a, φ(b) = b
11	26531	⟨a, b \| aa=1, aabbaaab=b⟩	φ(a) = a, φ(b) = b
11	26645	⟨a, b \| aa=1, baaaabab=b⟩	φ(a) = a, φ(b) = b
11	27019	⟨a, b \| aa=1, aaabbab=ab⟩	φ(a) = a, φ(b) = b
11	27063	⟨a, b \| aa=1, aabbaba=ba⟩	φ(a) = a, φ(b) = b
11	27096	⟨a, b \| aa=1, abaabab=ab⟩	φ(a) = a, φ(b) = b
11	27131	⟨a, b \| aa=1, abbaaab=ab⟩	φ(a) = a, φ(b) = b
11	27589	⟨a, b \| aa=1, aabbab=aab⟩	φ(a) = a, φ(b) = b
11	27592	⟨a, b \| aa=1, aabbab=baa⟩	φ(a) = a, φ(b) = b
11	27711	⟨a, b \| aa=1, baabab=aab⟩	φ(a) = a, φ(b) = b
11	27714	⟨a, b \| aa=1, baabab=baa⟩	φ(a) = a, φ(b) = b
11	28047	⟨a, b \| aa=1, aaaab=bbab⟩	φ(a) = a, φ(b) = b
11	28183	⟨a, b \| aa=1, abbab=aaab⟩	φ(a) = a, φ(b) = b
11	28186	⟨a, b \| aa=1, abbab=abaa⟩	φ(a) = a, φ(b) = b