#5675 ⟨a, b, c | aa=b, aac=ca⟩

Properties

Sum of relation sides is 8
Infinite cancellative non-commutative monoid
Element a has infinite order
Submonoid of enveloping group: ⟨a, b | aaba^-1b^-1⟩
- Embedding: φ(a) = a, φ(b) = aa, φ(c) = b

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(c) = 0, a < c; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a²c ⇒ ca	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,aac=ca ac/b - -
aac=ca
b=aa

6 unique, 25 total

Σ	#	Presentation	Properties	φ
7	417	⟨a, b, c \| abc=b, cba=1⟩	Grp Inf	10
7	537	⟨a, b, c \| ba=ac, bb=c⟩	Can Inf	4
7	554	⟨a, b, c \| bb=ac, ca=b⟩	Can Inf	3
7	1041	⟨a, b, c \| ab=c, bca=c⟩	Can Inf	2
8	5567	⟨a, b, c \| ab=c, baca=c⟩	Can Inf
8	6101	⟨a, b, c \| ab=c, cac=ba⟩	Can Inf

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

1 total

Σ	#	Presentation	Mapping
8	5999	⟨a, b, c \| ab=c, aab=ba⟩	φ(a) = a, φ(b) = c, φ(c) = ac