#173 ⟨a, b, c | aa=1, bbcc=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 ⋅ bc²b = b and b ⋅ 1 = b, however bc²b ≠ 1
Not right cancellative, because right multiplication by bc² is not injective:
- bc²b ⋅ bc² = bc² and 1 ⋅ bc² = bc², however bc²b ≠ 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.	b²c² ⇒ 1	[2]

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

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

7 total

Σ	#	Presentation	Mapping
8	2441	⟨a, b, c \| aab=b, bbcc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6128	⟨a, b, c \| aa=1, aabbcc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6170	⟨a, b, c \| aa=1, abbcca=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6200	⟨a, b, c \| aa=1, baabcc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6236	⟨a, b, c \| aa=1, bbaacc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	6672	⟨a, b, c \| aa=1, abbcc=a⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	7259	⟨a, b, c \| aa=1, bbcc=aa⟩	φ(a) = a, φ(b) = b, φ(c) = c