#144 ⟨a, b, c | aa=b, bc=b⟩

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 ⋅ ac = 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.	a²c ⇒ a²	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,bc=b ac/b - -
aac=aa
b=aa

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

7 total

Σ	#	Presentation	Mapping
7	504	⟨a, b, c \| ab=aa, ac=b⟩	φ(a) = a, φ(b) = ac, φ(c) = c
7	904	⟨a, b, c \| aa=b, aac=b⟩	φ(a) = a, φ(b) = aa, φ(c) = c
7	1052	⟨a, b, c \| aa=b, bc=aa⟩	φ(a) = a, φ(b) = aa, φ(c) = c
7	1058	⟨a, b, c \| ab=a, cc=ab⟩	φ(a) = aa, φ(b) = c, φ(c) = a
8	5669	⟨a, b, c \| aa=b, aac=aa⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5804	⟨a, b, c \| ab=a, abb=cc⟩	φ(a) = aa, φ(b) = c, φ(c) = a
8	5996	⟨a, b, c \| ab=c, aab=aa⟩	φ(a) = a, φ(b) = c, φ(c) = ac