#2826 ⟨a, b, c | aaa=b, aca=b⟩

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 ⋅ ca = a³ and a ⋅ a² = a³, however ca ≠ a²
Not right cancellative, because right multiplication by a is not injective:
- ac ⋅ a = a³ and a² ⋅ a = a³, however ac ≠ a²

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.	aca ⇒ a³	[2]
2.	b ⇒ a³	[1]

# abc:aaa=b,aca=b ac/b - -
aca=aaa
b=aaa

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

2 total

Σ	#	Presentation	Mapping
8	5696	⟨a, b, c \| aa=b, aca=ab⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5993	⟨a, b, c \| ab=c, aaa=ca⟩	φ(a) = a, φ(b) = c, φ(c) = ac