#3623 ⟨a, b, c | aaa=1, abcbc=1⟩

Properties

Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element b has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ (cb)⁶ = b and b ⋅ 1 = b, however (cb)⁶ ≠ 1
Not right cancellative, because right multiplication by c(bc)⁵ is not injective:
- (cb)⁶ ⋅ c(bc)⁵ = c(bc)⁵ and 1 ⋅ c(bc)⁵ = c(bc)⁵, however (cb)⁶ ≠ 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)⁶ ⇒ 1	[6]
2.	a ⇒ (bc)⁴	[5]

# abc:aaa=1,abcbc=1 bc/a - -
bcbcbcbcbcbc=1
a=bcbcbcbc

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

2 total

Σ	#	Presentation	Mapping
8	3648	⟨a, b, c \| aaa=1, bcabc=1⟩	φ(a) = bcbcbcbc, φ(b) = b, φ(c) = c
8	4150	⟨a, b, c \| aaa=1, bcbc=a⟩	φ(a) = bcbc, φ(b) = b, φ(c) = c