#136 ⟨a, b, c | aa=a, bc=a⟩

Properties

Sum of relation sides is 6
Infinite non-cancellative non-commutative monoid
Element b has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ cbc = bc and b ⋅ c = bc, however cbc ≠ c
Not right cancellative, because right multiplication by c is not injective:
- bcb ⋅ c = bc and b ⋅ c = bc, however bcb ≠ b

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)² ⇒ bc	[4]
2.	a ⇒ bc	[2]

# abc:aa=a,bc=a bc/a - -
bcbc=bc
a=bc

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

9 total

Σ	#	Presentation	Mapping
7	1022	⟨a, b, c \| ab=c, abc=c⟩	φ(a) = b, φ(b) = c, φ(c) = bc
7	1050	⟨a, b, c \| aa=a, bc=aa⟩	φ(a) = bc, φ(b) = b, φ(c) = c
7	1064	⟨a, b, c \| ab=c, cc=ab⟩	φ(a) = b, φ(b) = c, φ(c) = bc
8	3391	⟨a, b, c \| ac=ab, bab=c⟩	φ(a) = b, φ(b) = c, φ(c) = cbc
8	3394	⟨a, b, c \| ac=ab, bac=c⟩	φ(a) = b, φ(b) = c, φ(c) = cbc
8	5515	⟨a, b, c \| ab=c, abab=c⟩	φ(a) = b, φ(b) = c, φ(c) = bc
8	5611	⟨a, b, c \| aa=a, aaa=bc⟩	φ(a) = bc, φ(b) = b, φ(c) = c
8	6024	⟨a, b, c \| ab=c, abc=ab⟩	φ(a) = b, φ(b) = c, φ(c) = bc
8	6086	⟨a, b, c \| ab=c, bca=ba⟩	φ(a) = c, φ(b) = b, φ(c) = cb