#2341 ⟨a, b, c | aaa=b, bccc=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 ⋅ a²c³a = a and a ⋅ 1 = a, however a²c³a ≠ 1
Not right cancellative, because right multiplication by a²c³ is not injective:
- a²c³a ⋅ a²c³ = a²c³ and 1 ⋅ a²c³ = a²c³, however a²c³a ≠ 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.	a³c³ ⇒ 1	[2]
2.	b ⇒ a³	[1]

# abc:aaa=b,bccc=1 ac/b - -
aaaccc=1
b=aaa

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

7 total

Σ	#	Presentation	Mapping
8	2500	⟨a, b, c \| aab=c, acbb=1⟩	φ(a) = a, φ(b) = c, φ(c) = aac
8	4218	⟨a, b, c \| aab=1, accc=b⟩	φ(a) = a, φ(b) = accc, φ(c) = c
8	4506	⟨a, b, c \| abc=1, aacc=b⟩	φ(a) = a, φ(b) = aacc, φ(c) = c
8	4693	⟨a, b, c \| aa=b, abccc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4728	⟨a, b, c \| aa=b, baccc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5004	⟨a, b, c \| ab=c, aacbb=1⟩	φ(a) = a, φ(b) = c, φ(c) = ac
8	6841	⟨a, b, c \| ab=1, aaccc=b⟩	φ(a) = a, φ(b) = aaccc, φ(c) = c