#168 ⟨a, b, c | aa=1, babc=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 ⋅ abcb = b and b ⋅ 1 = b, however abcb ≠ 1
Not right cancellative, because right multiplication by abc is not injective:
- abcb ⋅ abc = abc and 1 ⋅ abc = abc, however abcb ≠ 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.	babc ⇒ 1	[2]

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

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

6 total

Σ	#	Presentation	Mapping
8	2433	⟨a, b, c \| aab=b, babc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6120	⟨a, b, c \| aa=1, aababc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6145	⟨a, b, c \| aa=1, ababca=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6194	⟨a, b, c \| aa=1, baaabc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6646	⟨a, b, c \| aa=1, ababc=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7211	⟨a, b, c \| aa=1, babc=aa⟩	φ(a) = a, φ(b) = b, φ(c) = c