#1760 ⟨a, b, c | aab=cb, bcc=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 b is not injective:
- b ⋅ a⁴b = b and b ⋅ 1 = b, however a⁴b ≠ 1
Not right cancellative, because right multiplication by a⁴ is not injective:
- a⁴b ⋅ a⁴ = a⁴ and 1 ⋅ a⁴ = a⁴, however a⁴b ≠ 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.	ba⁴ ⇒ 1	[4]
2.	c ⇒ a²	[3]

# abc:aab=cb,bcc=1 ab/c - -
baaaa=1
c=aa

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

3 total

Σ	#	Presentation	Mapping
8	4714	⟨a, b, c \| aa=b, acccc=1⟩	φ(a) = b, φ(b) = bb, φ(c) = a
8	4902	⟨a, b, c \| ab=a, caaaa=1⟩	φ(a) = a, φ(b) = 1, φ(c) = b
8	5052	⟨a, b, c \| ab=c, baaaa=1⟩	φ(a) = a, φ(b) = b, φ(c) = ab