#939 ⟨a, b, c | aa=b, cbc=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 c is not injective:
- c ⋅ a²c = c and c ⋅ 1 = c, however a²c ≠ 1
Not right cancellative, because right multiplication by c is not injective:
- ca² ⋅ c = c and 1 ⋅ c = c, however ca² ≠ 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.	ca²c ⇒ c	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,cbc=c ac/b - -
caac=c
b=aa

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

6 total

Σ	#	Presentation	Mapping
7	1000	⟨a, b, c \| ab=a, cca=b⟩	φ(a) = c, φ(b) = aac, φ(c) = a
7	1035	⟨a, b, c \| ab=c, bac=b⟩	φ(a) = a, φ(b) = c, φ(c) = ac
8	5208	⟨a, b, c \| aa=b, acca=a⟩	φ(a) = c, φ(b) = cc, φ(c) = a
8	5258	⟨a, b, c \| aa=b, caac=c⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5462	⟨a, b, c \| ab=a, ccab=b⟩	φ(a) = c, φ(b) = aac, φ(c) = a
8	5519	⟨a, b, c \| ab=c, abba=a⟩	φ(a) = c, φ(b) = a, φ(c) = ca