#20 ⟨a, b, c | aa=1, bbc=1⟩

Properties

Sum of relation sides is 5
Infinite non-cancellative non-commutative monoid
Element ba has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ bcb = b and b ⋅ 1 = b, however bcb ≠ 1
Not right cancellative, because right multiplication by bc is not injective:
- bcb ⋅ bc = bc and 1 ⋅ bc = bc, however bcb ≠ 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.	b²c ⇒ 1	[2]

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

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

9 total

Σ	#	Presentation	Mapping
7	306	⟨a, b, c \| aab=b, bbc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1067	⟨a, b, c \| aa=1, aabbc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1078	⟨a, b, c \| aa=1, abbca=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1089	⟨a, b, c \| aa=1, baabc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1237	⟨a, b, c \| aa=1, abbc=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
7	1420	⟨a, b, c \| aa=1, bbc=aa⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	3662	⟨a, b, c \| aab=1, aabcc=1⟩	φ(a) = b, φ(b) = c, φ(c) = a
8	3788	⟨a, b, c \| aab=1, caabc=1⟩	φ(a) = b, φ(b) = c, φ(c) = a
8	3838	⟨a, b, c \| aab=1, ccaab=1⟩	φ(a) = b, φ(b) = c, φ(c) = a