#229 ⟨a, b, c | ab=1, aac=a⟩

Properties

Sum of relation sides is 6
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 b is not injective:
- ba ⋅ b = b and 1 ⋅ b = b, however ba ≠ 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.	ab ⇒ 1	[1]
2.	a²c ⇒ a	[2]

# abc:ab=1,aac=a abc - -
ab=1
aac=a

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

9 total

Σ	#	Presentation	Mapping
8	2349	⟨a, b, c \| aab=a, aabc=1⟩	φ(a) = a, φ(b) = c, φ(c) = b
8	6787	⟨a, b, c \| ab=1, aaabc=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6801	⟨a, b, c \| ab=1, aabac=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6820	⟨a, b, c \| ab=1, aacab=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6845	⟨a, b, c \| ab=1, abaac=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7313	⟨a, b, c \| ab=1, aaca=aa⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7466	⟨a, b, c \| ab=1, baac=ba⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7708	⟨a, b, c \| ab=1, aac=aab⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7709	⟨a, b, c \| ab=1, aba=aac⟩	φ(a) = a, φ(b) = b, φ(c) = c