#2094 ⟨a, b | aba=a, abba=a⟩

Properties

Sum of relation sides is 9
Infinite non-cancellative non-commutative monoid
Element a 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.	ab²a ⇒ a	[2]

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

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

4 total

Σ	#	Presentation	Mapping
11	12518	⟨a, b \| abab=ab, abba=a⟩	φ(a) = a, φ(b) = b
11	19126	⟨a, b \| aba=a, ababba=a⟩	φ(a) = a, φ(b) = b
11	19674	⟨a, b \| aba=a, abbab=ab⟩	φ(a) = a, φ(b) = b
11	20216	⟨a, b \| aba=a, abba=aba⟩	φ(a) = a, φ(b) = b