#942 ⟨a, b, c | aa=b, ccc=c⟩

Properties

Sum of relation sides is 7
Isomorphic to ℕ_{(3 = 1)} ∗ ℕ
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by c is not injective:
- c ⋅ c² = c and c ⋅ 1 = c, however c² ≠ 1
Not right cancellative, because right multiplication by c is not injective:
- c² ⋅ c = c and 1 ⋅ c = c, however c² ≠ 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

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.	c³ ⇒ c	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,ccc=c ac/b - -
ccc=c
b=aa

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

6 total

Σ	#	Presentation	Mapping
7	1008	⟨a, b, c \| ab=c, aaa=a⟩	φ(a) = c, φ(b) = a, φ(c) = ca
8	2800	⟨a, b, c \| aaa=a, aab=c⟩	φ(a) = c, φ(b) = a, φ(c) = cca
8	2801	⟨a, b, c \| aaa=a, aba=c⟩	φ(a) = c, φ(b) = a, φ(c) = cac
8	2802	⟨a, b, c \| aaa=a, abb=c⟩	φ(a) = c, φ(b) = a, φ(c) = caa
8	2806	⟨a, b, c \| aaa=a, bab=c⟩	φ(a) = c, φ(b) = a, φ(c) = aca
8	2809	⟨a, b, c \| aaa=a, bbb=c⟩	φ(a) = c, φ(b) = a, φ(c) = aaa