#863 ⟨a, b, c | ab=c, accc=1⟩

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 ⋅ a(ba)³ = a and a ⋅ 1 = a, however a(ba)³ ≠ 1
Not right cancellative, because right multiplication by (ab)³ is not injective:
- a(ba)³ ⋅ (ab)³ = (ab)³ and 1 ⋅ (ab)³ = (ab)³, however a(ba)³ ≠ 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(b) = 0, a < b; deg(c) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a(ab)³ ⇒ 1	[2]
2.	c ⇒ ab	[1]

# abc:ab=c,accc=1 ab/c - -
aababab=1
c=ab

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

3 total

Σ	#	Presentation	Mapping
8	4999	⟨a, b, c \| ab=c, aabcc=1⟩	φ(a) = a, φ(b) = b, φ(c) = ab
8	5035	⟨a, b, c \| ab=c, acabc=1⟩	φ(a) = a, φ(b) = b, φ(c) = ab
8	5081	⟨a, b, c \| ab=c, bbcca=1⟩	φ(a) = b, φ(b) = a, φ(c) = ba