#176 ⟨a, b, c | ab=1, aaac=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.	a³c ⇒ 1	[2]

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

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

10 total

Σ	#	Presentation	Mapping
8	2559	⟨a, b, c \| aba=a, aaac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	2713	⟨a, b, c \| abc=a, bbba=1⟩	φ(a) = c, φ(b) = a, φ(c) = b
8	6274	⟨a, b, c \| ab=1, aaaabc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6278	⟨a, b, c \| ab=1, aaabac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6284	⟨a, b, c \| ab=1, aaacab=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6292	⟨a, b, c \| ab=1, aabaac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6336	⟨a, b, c \| ab=1, abaaac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6790	⟨a, b, c \| ab=1, aaaca=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6962	⟨a, b, c \| ab=1, baaac=b⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7288	⟨a, b, c \| ab=1, aaac=ab⟩	φ(a) = a, φ(b) = b, φ(c) = c