#680 ⟨a, b, c | abc=1, aaaa=1⟩

Properties

Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element b has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ (cb)⁴ = b and b ⋅ 1 = b, however (cb)⁴ ≠ 1
Not right cancellative, because right multiplication by c(bc)³ is not injective:
- (cb)⁴ ⋅ c(bc)³ = c(bc)³ and 1 ⋅ c(bc)³ = c(bc)³, however (cb)⁴ ≠ 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(b) = deg(c) = 0, b < c; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	(bc)⁴ ⇒ 1	[7]
2.	a ⇒ (bc)³	[6]

# abc:abc=1,aaaa=1 bc/a - -
bcbcbcbc=1
a=bcbcbc

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

9 total

Σ	#	Presentation	Mapping
7	884	⟨a, b, c \| ab=c, cccc=1⟩	φ(a) = b, φ(b) = c, φ(c) = bc
7	1085	⟨a, b, c \| aa=1, abcbc=1⟩	φ(a) = bcbc, φ(b) = b, φ(c) = c
7	1110	⟨a, b, c \| aa=1, bcabc=1⟩	φ(a) = bcbc, φ(b) = b, φ(c) = c
7	1270	⟨a, b, c \| aa=1, bcbc=a⟩	φ(a) = bcbc, φ(b) = b, φ(c) = c
8	1584	⟨a, b, c \| aaa=bc, abc=1⟩	φ(a) = bcbcbc, φ(b) = b, φ(c) = c
8	2665	⟨a, b, c \| aba=c, bcbc=1⟩	φ(a) = c, φ(b) = b, φ(c) = cbc
8	5031	⟨a, b, c \| ab=c, abccc=1⟩	φ(a) = b, φ(b) = c, φ(c) = bc
8	5092	⟨a, b, c \| ab=c, bccca=1⟩	φ(a) = c, φ(b) = b, φ(c) = cb
8	5098	⟨a, b, c \| ab=c, cabcc=1⟩	φ(a) = b, φ(b) = c, φ(c) = bc