#140 ⟨a, b, c | aa=b, ac=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 ⋅ c = a² and a ⋅ a = a², however c ≠ 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(c) = 0, a < c; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	ac ⇒ a²	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,ac=b ac/b - -
ac=aa
b=aa

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

11 total

Σ	#	Presentation	Mapping
7	506	⟨a, b, c \| ab=aa, ba=c⟩	φ(a) = a, φ(b) = c, φ(c) = ca
7	507	⟨a, b, c \| ab=aa, bb=c⟩	φ(a) = a, φ(b) = c, φ(c) = cc
7	1051	⟨a, b, c \| aa=b, ac=aa⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	3307	⟨a, b, c \| ab=aa, aaa=c⟩	φ(a) = a, φ(b) = c, φ(c) = aaa
8	3308	⟨a, b, c \| ab=aa, aab=c⟩	φ(a) = a, φ(b) = c, φ(c) = aaa
8	3312	⟨a, b, c \| ab=aa, aba=c⟩	φ(a) = a, φ(b) = c, φ(c) = aaa
8	3313	⟨a, b, c \| ab=aa, abb=c⟩	φ(a) = a, φ(b) = c, φ(c) = aaa
8	3326	⟨a, b, c \| ab=aa, baa=c⟩	φ(a) = a, φ(b) = c, φ(c) = caa
8	3327	⟨a, b, c \| ab=aa, bab=c⟩	φ(a) = a, φ(b) = c, φ(c) = caa
8	3331	⟨a, b, c \| ab=aa, bba=c⟩	φ(a) = a, φ(b) = c, φ(c) = cca
8	3332	⟨a, b, c \| ab=aa, bbb=c⟩	φ(a) = a, φ(b) = c, φ(c) = ccc