#174 ⟨a, b, c | aa=1, bcbc=1⟩

Properties

Sum of relation sides is 6
Infinite non-cancellative non-commutative monoid
Element ba has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ (cb)² = b and b ⋅ 1 = b, however (cb)² ≠ 1
Not right cancellative, because right multiplication by cbc is not injective:
- (cb)² ⋅ cbc = cbc and 1 ⋅ cbc = cbc, however (cb)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

#	Rule	Proof
1.	a² ⇒ 1	[1]
2.	(bc)² ⇒ 1	[2]

# abc:aa=1,bcbc=1 abc - -
aa=1
bcbc=1

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

7 total

Σ	#	Presentation	Mapping
8	2447	⟨a, b, c \| aab=b, bcbc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6134	⟨a, b, c \| aa=1, aabcbc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6183	⟨a, b, c \| aa=1, abcbca=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6204	⟨a, b, c \| aa=1, baacbc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6261	⟨a, b, c \| aa=1, bcaabc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6687	⟨a, b, c \| aa=1, abcbc=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7265	⟨a, b, c \| aa=1, bcbc=aa⟩	φ(a) = a, φ(b) = b, φ(c) = c