#100 ⟨a, b, c | ab=a, aac=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 ⋅ b = a and a ⋅ 1 = a, however b ≠ 1
Not right cancellative, because right multiplication by ac is not injective:
- aca ⋅ ac = ac and 1 ⋅ ac = ac, however aca ≠ 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 ⇒ a	[1]
2.	a²c ⇒ 1	[2]

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

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

6 total

Σ	#	Presentation	Mapping
7	776	⟨a, b, c \| ab=a, aabc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	780	⟨a, b, c \| ab=a, abac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	1936	⟨a, b, c \| abc=ab, bba=1⟩	φ(a) = c, φ(b) = a, φ(c) = b
8	4778	⟨a, b, c \| ab=a, aabbc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	4792	⟨a, b, c \| ab=a, ababc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	4796	⟨a, b, c \| ab=a, abbac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c