#231 ⟨a, b, c | ab=1, aac=c⟩

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.	a²c ⇒ c	[2]

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

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

8 total

Σ	#	Presentation	Mapping
8	6789	⟨a, b, c \| ab=1, aaabc=c⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6803	⟨a, b, c \| ab=1, aabac=c⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6822	⟨a, b, c \| ab=1, aacab=c⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6847	⟨a, b, c \| ab=1, abaac=c⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7319	⟨a, b, c \| ab=1, aaca=ca⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7468	⟨a, b, c \| ab=1, baac=bc⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7713	⟨a, b, c \| ab=1, abc=aac⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7820	⟨a, b, c \| ab=1, cab=aac⟩	φ(a) = a, φ(b) = b, φ(c) = c