#22 ⟨a, b, c | ab=1, aac=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.	a²c ⇒ 1	[2]

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

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

11 total

Σ	#	Presentation	Mapping
7	391	⟨a, b, c \| abc=a, bba=1⟩	φ(a) = c, φ(b) = a, φ(c) = b
7	1116	⟨a, b, c \| ab=1, aaabc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1120	⟨a, b, c \| ab=1, aabac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1126	⟨a, b, c \| ab=1, aacab=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1134	⟨a, b, c \| ab=1, abaac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1285	⟨a, b, c \| ab=1, aaca=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1334	⟨a, b, c \| ab=1, baac=b⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1446	⟨a, b, c \| ab=1, aac=ab⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	3654	⟨a, b, c \| aab=1, aaabc=1⟩	φ(a) = a, φ(b) = c, φ(c) = b
8	3658	⟨a, b, c \| aab=1, aabac=1⟩	φ(a) = a, φ(b) = c, φ(c) = b
8	3692	⟨a, b, c \| aab=1, acaab=1⟩	φ(a) = a, φ(b) = c, φ(c) = b