#1043 ⟨a, b, c | ab=c, bcc=b⟩

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 b is not injective:
- b ⋅ (ab)² = b and b ⋅ 1 = b, however (ab)² ≠ 1
Not right cancellative, because right multiplication by b is not injective:
- (ba)² ⋅ b = b and 1 ⋅ b = b, however (ba)² ≠ 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(b) = 0, a < b; deg(c) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b(ab)² ⇒ b	[2]
2.	c ⇒ ab	[1]

# abc:ab=c,bcc=b ab/c - -
babab=b
c=ab

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

5 total

Σ	#	Presentation	Mapping
8	3023	⟨a, b, c \| aba=a, cac=b⟩	φ(a) = b, φ(b) = aba, φ(c) = a
8	3044	⟨a, b, c \| aba=c, abc=a⟩	φ(a) = b, φ(b) = a, φ(c) = bab
8	5423	⟨a, b, c \| ab=a, caca=b⟩	φ(a) = b, φ(b) = abab, φ(c) = a
8	5525	⟨a, b, c \| ab=c, abca=a⟩	φ(a) = b, φ(b) = a, φ(c) = ba
8	5563	⟨a, b, c \| ab=c, babc=b⟩	φ(a) = a, φ(b) = b, φ(c) = ab