#512 ⟨a, b, c | ab=aa, ca=b⟩

Properties

Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ ca = a² and a ⋅ a = a², however ca ≠ a
Not right cancellative, because right multiplication by a is not injective:
- ac ⋅ a = a² and a ⋅ a = a², however ac ≠ a

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

Reduction order:
- Left-to-right recursive path with deg(a) = deg(c) = 0, a < c; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	aca ⇒ a²	[3]
2.	b ⇒ ca	[2]

# abc:ab=aa,ca=b ac/b - -
aca=aa
b=ca

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

4 total

Σ	#	Presentation	Mapping
7	912	⟨a, b, c \| aa=b, aca=b⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5695	⟨a, b, c \| aa=b, aca=aa⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5763	⟨a, b, c \| aa=b, cac=cc⟩	φ(a) = c, φ(b) = cc, φ(c) = a
8	6014	⟨a, b, c \| ab=c, aba=aa⟩	φ(a) = a, φ(b) = c, φ(c) = ac