#4737 ⟨a, b, c | aa=b, bbcac=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 ⋅ a³(ca)² = a and a ⋅ 1 = a, however a³(ca)² ≠ 1
Not right cancellative, because right multiplication by a³cac is not injective:
- a³(ca)² ⋅ a³cac = a³cac and 1 ⋅ a³cac = a³cac, however a³(ca)² ≠ 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.	a⁴cac ⇒ 1	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,bbcac=1 ac/b - -
aaaacac=1
b=aa

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

1 total

Σ	#	Presentation	Mapping
8	4991	⟨a, b, c \| ab=c, aaacc=1⟩	φ(a) = a, φ(b) = c, φ(c) = ac