#2828 ⟨a, b, c | aaa=b, acb=a⟩

Properties

Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ ca³ = a and a ⋅ 1 = a, however ca³ ≠ 1
Not right cancellative, because right multiplication by a is not injective:
- aca² ⋅ a = a and 1 ⋅ a = a, however aca² ≠ 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.	aca³ ⇒ a	[2]
2.	b ⇒ a³	[1]

# abc:aaa=b,acb=a ac/b - -
acaaa=a
b=aaa

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

2 total

Σ	#	Presentation	Mapping
8	5196	⟨a, b, c \| aa=b, acab=a⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5405	⟨a, b, c \| ab=a, caaa=b⟩	φ(a) = a, φ(b) = caaa, φ(c) = c

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

6 total

Σ	#	Presentation	Mapping
8	2864	⟨a, b, c \| aab=a, aca=b⟩	φ(a) = a, φ(b) = aca, φ(c) = c
8	2969	⟨a, b, c \| aab=c, aca=a⟩	φ(a) = a, φ(b) = c, φ(c) = aac
8	5187	⟨a, b, c \| aa=b, abca=a⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5285	⟨a, b, c \| ab=a, aaca=b⟩	φ(a) = a, φ(b) = aaca, φ(c) = c
8	5502	⟨a, b, c \| ab=c, aaca=a⟩	φ(a) = a, φ(b) = c, φ(c) = ac
8	5577	⟨a, b, c \| ab=c, bbbc=b⟩	φ(a) = c, φ(b) = a, φ(c) = ca