#189 ⟨a, b, c | ab=1, acbc=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.	acbc ⇒ 1	[2]

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

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

9 total

Σ	#	Presentation	Mapping
8	2572	⟨a, b, c \| aba=a, acbc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	2707	⟨a, b, c \| abc=a, baca=1⟩	φ(a) = c, φ(b) = a, φ(c) = b
8	6307	⟨a, b, c \| ab=1, aabcbc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6350	⟨a, b, c \| ab=1, abacbc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6404	⟨a, b, c \| ab=1, acabbc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6418	⟨a, b, c \| ab=1, acbabc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6934	⟨a, b, c \| ab=1, acbca=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6996	⟨a, b, c \| ab=1, bacbc=b⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7426	⟨a, b, c \| ab=1, acbc=ab⟩	φ(a) = a, φ(b) = b, φ(c) = c