#1016 ⟨a, b, c | ab=c, aac=c⟩

Properties

Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ a²b = ab and a ⋅ b = ab, however a²b ≠ b
Not right cancellative, because right multiplication by b is not injective:
- a³ ⋅ b = ab and a ⋅ b = ab, however a³ ≠ a

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(b) = 0, a < b; deg(c) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a³b ⇒ ab	[2]
2.	c ⇒ ab	[1]

# abc:ab=c,aac=c ab/c - -
aaab=ab
c=ab

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

6 total

Σ	#	Presentation	Mapping
8	3375	⟨a, b, c \| ac=ab, aab=b⟩	φ(a) = a, φ(b) = aab, φ(c) = b
8	3376	⟨a, b, c \| ac=ab, aab=c⟩	φ(a) = a, φ(b) = b, φ(c) = aab
8	5490	⟨a, b, c \| ab=c, aaab=c⟩	φ(a) = a, φ(b) = b, φ(c) = ab
8	5688	⟨a, b, c \| aa=b, abc=ac⟩	φ(a) = a, φ(b) = aa, φ(c) = b
8	5724	⟨a, b, c \| aa=b, bac=ac⟩	φ(a) = a, φ(b) = aa, φ(c) = b
8	6006	⟨a, b, c \| ab=c, aac=ab⟩	φ(a) = a, φ(b) = b, φ(c) = ab