#180 ⟨a, b, c | ab=1, aacc=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.	a²c² ⇒ 1	[2]

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

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

10 total

Σ	#	Presentation	Mapping
8	2563	⟨a, b, c \| aba=a, aacc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	2710	⟨a, b, c \| abc=a, bbaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = b
8	6282	⟨a, b, c \| ab=1, aaabcc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6296	⟨a, b, c \| ab=1, aabacc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6315	⟨a, b, c \| ab=1, aacabc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6328	⟨a, b, c \| ab=1, aaccab=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6340	⟨a, b, c \| ab=1, abaacc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6834	⟨a, b, c \| ab=1, aacca=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6972	⟨a, b, c \| ab=1, baacc=b⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7332	⟨a, b, c \| ab=1, aacc=ab⟩	φ(a) = a, φ(b) = b, φ(c) = c