#2084 ⟨a, b | aba=a, aaaa=a⟩

Properties

Sum of relation sides is 9
Infinite non-cancellative non-commutative monoid
Element a³b² has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ ba = a and a ⋅ 1 = a, however ba ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- ab ⋅ a = a and 1 ⋅ a = a, however ab ≠ 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.	aba ⇒ a	[1]
2.	a⁴ ⇒ a	[2]

# ab:aba=a,aaaa=a ab - -
aba=a
aaaa=a

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

5 total

Σ	#	Presentation	Mapping
11	12639	⟨a, b \| babb=bb, bbbb=b⟩	φ(a) = b, φ(b) = a
11	19092	⟨a, b \| aba=a, aaaaba=a⟩	φ(a) = a, φ(b) = b
11	19096	⟨a, b \| aba=a, aaabaa=a⟩	φ(a) = a, φ(b) = b
11	19636	⟨a, b \| aba=a, aaaab=ab⟩	φ(a) = a, φ(b) = b
11	20178	⟨a, b \| aba=a, aaaa=aba⟩	φ(a) = a, φ(b) = b