#187 ⟨a, b, c | ab=1, acac=1⟩

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.	(ac)² ⇒ 1	[2]

# abc:ab=1,acac=1 abc - -
ab=1
acac=1

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

9 total

Σ	#	Presentation	Mapping
8	2570	⟨a, b, c \| aba=a, acac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	2704	⟨a, b, c \| abc=a, baba=1⟩	φ(a) = c, φ(b) = a, φ(c) = b
8	6304	⟨a, b, c \| ab=1, aabcac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6348	⟨a, b, c \| ab=1, abacac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6397	⟨a, b, c \| ab=1, acaabc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6402	⟨a, b, c \| ab=1, acabac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6913	⟨a, b, c \| ab=1, acaca=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6991	⟨a, b, c \| ab=1, bacac=b⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7408	⟨a, b, c \| ab=1, acac=ab⟩	φ(a) = a, φ(b) = b, φ(c) = c