#4685 ⟨a, b, c | aa=b, abbcc=1⟩

Properties

Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ a⁴c²a = a and a ⋅ 1 = a, however a⁴c²a ≠ 1
Not right cancellative, because right multiplication by a⁴c² is not injective:
- a⁴c²a ⋅ a⁴c² = a⁴c² and 1 ⋅ a⁴c² = a⁴c², however a⁴c²a ≠ 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.	a⁵c² ⇒ 1	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,abbcc=1 ac/b - -
aaaaacc=1
b=aa

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

2 total

Σ	#	Presentation	Mapping
8	4722	⟨a, b, c \| aa=b, babcc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4732	⟨a, b, c \| aa=b, bbacc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c