#32 ⟨a, b, c | ab=1, ccc=1⟩

Properties

Sum of relation sides is 5
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.	c³ ⇒ 1	[2]

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

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

8 total

Σ	#	Presentation	Mapping
7	424	⟨a, b, c \| abc=c, ccc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1151	⟨a, b, c \| ab=1, abccc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1214	⟨a, b, c \| ab=1, cabcc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1368	⟨a, b, c \| ab=1, bccc=b⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1540	⟨a, b, c \| ab=1, ccc=ab⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	3603	⟨a, b, c \| aaa=1, aaabc=1⟩	φ(a) = c, φ(b) = a, φ(c) = b
8	3606	⟨a, b, c \| aaa=1, aabca=1⟩	φ(a) = c, φ(b) = a, φ(c) = b
8	3626	⟨a, b, c \| aaa=1, baaac=1⟩	φ(a) = c, φ(b) = a, φ(c) = b