#2545 ⟨a, b, c | aab=c, cbbb=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 ⋅ ab⁴a = a and a ⋅ 1 = a, however ab⁴a ≠ 1
Not right cancellative, because right multiplication by ab⁴ is not injective:
- ab⁴a ⋅ ab⁴ = ab⁴ and 1 ⋅ ab⁴ = ab⁴, however ab⁴a ≠ 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

Reduction order:
- Left-to-right recursive path with deg(a) = deg(b) = 0, a < b; deg(c) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a²b⁴ ⇒ 1	[2]
2.	c ⇒ a²b	[1]

# abc:aab=c,cbbb=1 ab/c - -
aabbbb=1
c=aab

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

3 total

Σ	#	Presentation	Mapping
8	4364	⟨a, b, c \| aab=1, cccc=b⟩	φ(a) = a, φ(b) = bbbb, φ(c) = b
8	4753	⟨a, b, c \| aa=b, bcccc=1⟩	φ(a) = a, φ(b) = aa, φ(c) = b
8	6958	⟨a, b, c \| ab=1, acccc=b⟩	φ(a) = a, φ(b) = abbbb, φ(c) = b