#2234 ⟨a, b, c | abab=1, abac=1⟩

Properties

Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ c = ab and a ⋅ b = ab, however c ≠ b
Not right cancellative, because right multiplication by bab is not injective:
- (ba)² ⋅ bab = bab and 1 ⋅ bab = bab, however (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

#	Rule	Proof
1.	ac ⇒ ab	[3]
2.	(ab)² ⇒ 1	[1]

# abc:abab=1,abac=1 abc - -
ac=ab
abab=1

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

2 total

Σ	#	Presentation	Mapping
8	3144	⟨a, b, c \| ac=ab, abab=1⟩	φ(a) = a, φ(b) = b, φ(c) = c
8	3145	⟨a, b, c \| ac=ab, abac=1⟩	φ(a) = a, φ(b) = b, φ(c) = c