#767 ⟨a, b | aab=1, abab=b⟩

Properties

Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ aba = a and a ⋅ 1 = a, however aba ≠ 1
Not right cancellative, because right multiplication by ab is not injective:
- aba ⋅ ab = ab and 1 ⋅ ab = ab, however aba ≠ 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.	a²b ⇒ 1	[1]
2.	bab ⇒ ab²	[3]

# ab:aab=1,abab=b ab - -
aab=1
bab=abb

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

13 total

Σ	#	Presentation	Mapping
10	7866	⟨a, b \| aab=1, ababa=ba⟩	φ(a) = a, φ(b) = b
10	7879	⟨a, b \| aab=1, abbab=bb⟩	φ(a) = a, φ(b) = b
10	7911	⟨a, b \| aab=1, babab=bb⟩	φ(a) = a, φ(b) = b
11	21819	⟨a, b \| aab=1, aaabbab=b⟩	φ(a) = a, φ(b) = b
11	21835	⟨a, b \| aab=1, aababab=b⟩	φ(a) = a, φ(b) = b
11	21863	⟨a, b \| aab=1, abaaabb=b⟩	φ(a) = a, φ(b) = b
11	21867	⟨a, b \| aab=1, abaabab=b⟩	φ(a) = a, φ(b) = b
11	21875	⟨a, b \| aab=1, ababaab=b⟩	φ(a) = a, φ(b) = b
11	23343	⟨a, b \| aab=1, abab=aabb⟩	φ(a) = a, φ(b) = b
11	23371	⟨a, b \| aab=1, baab=abab⟩	φ(a) = a, φ(b) = b
11	23381	⟨a, b \| aab=1, baba=abba⟩	φ(a) = a, φ(b) = b
11	23415	⟨a, b \| aab=1, bbab=abbb⟩	φ(a) = a, φ(b) = b
11	23419	⟨a, b \| aab=1, bbab=babb⟩	φ(a) = a, φ(b) = b