#3373 ⟨a, b, c | ac=ab, aaa=b⟩

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 ⋅ c = a⁴ and a ⋅ a³ = a⁴, however c ≠ a³

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) = 0; deg(b) = deg(c) = 1, b < c
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b ⇒ a³	[2]
2.	ac ⇒ a⁴	[3]

# abc:ac=ab,aaa=b a/bc - -
b=aaa
ac=aaaa

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

4 total

Σ	#	Presentation	Mapping
8	3436	⟨a, b, c \| ba=ac, aaa=b⟩	φ(a) = a, φ(b) = aaa, φ(c) = c
8	5487	⟨a, b, c \| ab=c, aaaa=c⟩	φ(a) = a, φ(b) = c, φ(c) = aaaa
8	5664	⟨a, b, c \| aa=b, aab=ac⟩	φ(a) = a, φ(b) = aa, φ(c) = c
8	5678	⟨a, b, c \| aa=b, aba=ac⟩	φ(a) = a, φ(b) = aa, φ(c) = c

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

1 total

Σ	#	Presentation	Mapping
8	5666	⟨a, b, c \| aa=b, aab=ca⟩	φ(a) = a, φ(b) = aa, φ(c) = c