#764 ⟨a, b, c | aa=b, bbcc=1⟩

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 ⋅ a³c²a = a and a ⋅ 1 = a, however a³c²a ≠ 1
Not right cancellative, because right multiplication by a³c² is not injective:
- a³c²a ⋅ a³c² = a³c² and 1 ⋅ a³c² = a³c², however a³c²a ≠ 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.	a⁴c² ⇒ 1	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,bbcc=1 ac/b - -
aaaacc=1
b=aa

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

10 total

Σ	#	Presentation	Mapping
8	2322	⟨a, b, c \| aaa=b, abcc=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2333	⟨a, b, c \| aaa=b, bacc=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2485	⟨a, b, c \| aab=c, aacb=1⟩	φ(a) = a, φ(b) = c, φ(c) = aac
8	4172	⟨a, b, c \| aab=1, aacc=b⟩	φ(a) = a, φ(b) = aacc, φ(c) = c
8	4488	⟨a, b, c \| abc=1, aaac=b⟩	φ(a) = a, φ(b) = aaac, φ(c) = c
8	4666	⟨a, b, c \| aa=b, aabcc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4680	⟨a, b, c \| aa=b, abacc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4718	⟨a, b, c \| aa=b, baacc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4990	⟨a, b, c \| ab=c, aaacb=1⟩	φ(a) = a, φ(b) = c, φ(c) = ac
8	6797	⟨a, b, c \| ab=1, aaacc=b⟩	φ(a) = a, φ(b) = aaacc, φ(c) = c