#2319 ⟨a, b, c | aaa=b, abbc=1⟩

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 ⋅ a⁶ca = a and a ⋅ 1 = a, however a⁶ca ≠ 1
Not right cancellative, because right multiplication by a⁶c is not injective:
- a⁶ca ⋅ a⁶c = a⁶c and 1 ⋅ a⁶c = a⁶c, however a⁶ca ≠ 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.	a⁷c ⇒ 1	[2]
2.	b ⇒ a³	[1]

# abc:aaa=b,abbc=1 ac/b - -
aaaaaaac=1
b=aaa

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

6 total

Σ	#	Presentation	Mapping
8	2331	⟨a, b, c \| aaa=b, babc=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	2334	⟨a, b, c \| aaa=b, bbac=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	4682	⟨a, b, c \| aa=b, abbbc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4720	⟨a, b, c \| aa=b, babbc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4730	⟨a, b, c \| aa=b, bbabc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	4733	⟨a, b, c \| aa=b, bbbac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = c