#126 ⟨a, b, c | ab=c, acc=1⟩

Properties

Sum of relation sides is 6
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ a(ba)² = a and a ⋅ 1 = a, however a(ba)² ≠ 1
Not right cancellative, because right multiplication by (ab)² is not injective:
- a(ba)² ⋅ (ab)² = (ab)² and 1 ⋅ (ab)² = (ab)², however a(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.	a(ab)² ⇒ 1	[2]
2.	c ⇒ ab	[1]

# abc:ab=c,acc=1 ab/c - -
aabab=1
c=ab

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

11 total

Σ	#	Presentation	Mapping
7	336	⟨a, b, c \| aab=c, cab=1⟩	φ(a) = a, φ(b) = b, φ(c) = aab
7	371	⟨a, b, c \| aba=c, acb=1⟩	φ(a) = a, φ(b) = b, φ(c) = aba
7	377	⟨a, b, c \| aba=c, bbc=1⟩	φ(a) = b, φ(b) = a, φ(c) = bab
7	765	⟨a, b, c \| aa=b, bcac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = b
7	845	⟨a, b, c \| ab=c, aabc=1⟩	φ(a) = a, φ(b) = b, φ(c) = ab
7	873	⟨a, b, c \| ab=c, bbca=1⟩	φ(a) = b, φ(b) = a, φ(c) = ba
7	1316	⟨a, b, c \| ab=1, acac=b⟩	φ(a) = a, φ(b) = abab, φ(c) = b
8	1912	⟨a, b, c \| abc=aa, baa=1⟩	φ(a) = ab, φ(b) = a, φ(c) = b
8	4669	⟨a, b, c \| aa=b, aacac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = b
8	4962	⟨a, b, c \| ab=a, ccaca=1⟩	φ(a) = b, φ(b) = 1, φ(c) = a
8	4993	⟨a, b, c \| ab=c, aabab=1⟩	φ(a) = a, φ(b) = b, φ(c) = ab