#155 ⟨a, b, c | ab=a, ca=b⟩

Properties

Sum of relation sides is 6
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ ca = a and a ⋅ 1 = a, however ca ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- ac ⋅ a = a and 1 ⋅ a = a, however ac ≠ 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.	aca ⇒ a	[3]
2.	b ⇒ ca	[2]

# abc:ab=a,ca=b ac/b - -
aca=a
b=ca

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

10 total

Σ	#	Presentation	Mapping
7	911	⟨a, b, c \| aa=b, aca=a⟩	φ(a) = a, φ(b) = aa, φ(c) = c
7	936	⟨a, b, c \| aa=b, cac=c⟩	φ(a) = c, φ(b) = cc, φ(c) = a
7	985	⟨a, b, c \| ab=a, cab=b⟩	φ(a) = a, φ(b) = ca, φ(c) = c
7	1017	⟨a, b, c \| ab=c, aba=a⟩	φ(a) = a, φ(b) = c, φ(c) = ac
8	2825	⟨a, b, c \| aaa=b, aca=a⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2850	⟨a, b, c \| aaa=b, cac=c⟩	φ(a) = c, φ(b) = ccc, φ(c) = a
8	2962	⟨a, b, c \| aab=c, aba=a⟩	φ(a) = a, φ(b) = c, φ(c) = aac
8	2978	⟨a, b, c \| aab=c, bab=b⟩	φ(a) = c, φ(b) = a, φ(c) = cca
8	3015	⟨a, b, c \| aba=a, bab=c⟩	φ(a) = a, φ(b) = c, φ(c) = cac
8	5417	⟨a, b, c \| ab=a, cabb=b⟩	φ(a) = a, φ(b) = ca, φ(c) = c