#277 ⟨a, b, c | aaa=b, bcc=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 ⋅ a²c²a = a and a ⋅ 1 = a, however a²c²a ≠ 1
Not right cancellative, because right multiplication by a²c² is not injective:
- a²c²a ⋅ a²c² = a²c² and 1 ⋅ a²c² = a²c², however a²c²a ≠ 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.	a³c² ⇒ 1	[2]
2.	b ⇒ a³	[1]

# abc:aaa=b,bcc=1 ac/b - -
aaacc=1
b=aaa

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

8 total

Σ	#	Presentation	Mapping
7	324	⟨a, b, c \| aab=c, acb=1⟩	φ(a) = a, φ(b) = c, φ(c) = aac
7	749	⟨a, b, c \| aa=b, abcc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
7	760	⟨a, b, c \| aa=b, bacc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
7	847	⟨a, b, c \| ab=c, aacb=1⟩	φ(a) = a, φ(b) = c, φ(c) = ac
7	1292	⟨a, b, c \| ab=1, aacc=b⟩	φ(a) = a, φ(b) = aacc, φ(c) = c
8	4661	⟨a, b, c \| aa=b, aaacc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4974	⟨a, b, c \| ab=a, cccaa=1⟩	φ(a) = c, φ(b) = 1, φ(c) = a
8	4987	⟨a, b, c \| ab=c, aaabb=1⟩	φ(a) = a, φ(b) = c, φ(c) = ac