#2853 ⟨a, b, c | aaa=b, cbc=c⟩

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 c is not injective:
- c ⋅ a³c = c and c ⋅ 1 = c, however a³c ≠ 1
Not right cancellative, because right multiplication by c is not injective:
- ca³ ⋅ c = c and 1 ⋅ c = c, however ca³ ≠ 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.	ca³c ⇒ c	[2]
2.	b ⇒ a³	[1]

# abc:aaa=b,cbc=c ac/b - -
caaac=c
b=aaa

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

4 total

Σ	#	Presentation	Mapping
8	2981	⟨a, b, c \| aab=c, bac=b⟩	φ(a) = a, φ(b) = c, φ(c) = aac
8	5261	⟨a, b, c \| aa=b, cabc=c⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5477	⟨a, b, c \| ab=a, ccca=b⟩	φ(a) = c, φ(b) = aaac, φ(c) = a
8	5558	⟨a, b, c \| ab=c, baac=b⟩	φ(a) = a, φ(b) = c, φ(c) = ac