#915 ⟨a, b, c | aa=b, acb=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 ⋅ ca² = a² and a ⋅ a = a², however ca² ≠ a
Not right cancellative, because right multiplication by a is not injective:
- aca ⋅ a = a² and a ⋅ a = a², however aca ≠ 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.	aca² ⇒ a²	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,acb=b ac/b - -
acaa=aa
b=aa

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

3 total

Σ	#	Presentation	Mapping
8	3346	⟨a, b, c \| ab=aa, caa=b⟩	φ(a) = a, φ(b) = caa, φ(c) = c
8	3349	⟨a, b, c \| ab=aa, cab=b⟩	φ(a) = a, φ(b) = caa, φ(c) = c
8	5701	⟨a, b, c \| aa=b, acb=aa⟩	φ(a) = a, φ(b) = aa, φ(c) = c

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

4 total

Σ	#	Presentation	Mapping
8	3318	⟨a, b, c \| ab=aa, aca=b⟩	φ(a) = a, φ(b) = aca, φ(c) = c
8	5170	⟨a, b, c \| aa=b, aaca=b⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	6032	⟨a, b, c \| ab=c, aca=aa⟩	φ(a) = a, φ(b) = c, φ(c) = ac
8	6078	⟨a, b, c \| ab=c, bbc=bb⟩	φ(a) = c, φ(b) = a, φ(c) = ca