#537 ⟨a, b, c | ba=ac, bb=c⟩

Properties

Sum of relation sides is 7
Infinite cancellative non-commutative monoid
Element a has infinite order
Submonoid of enveloping group: ⟨a, b | aaba^-1b^-1⟩
- Embedding: φ(a) = b^-1, φ(b) = b^-1ab, φ(c) = a

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.	ab² ⇒ ba	[3]
2.	c ⇒ b²	[2]

# abc:ba=ac,bb=c ab/c - -
abb=ba
c=bb

6 unique, 22 total

Σ	#	Presentation	Properties	φ
7	417	⟨a, b, c \| abc=b, cba=1⟩	Grp Inf	10
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	5675	⟨a, b, c \| aa=b, aac=ca⟩	Can Inf	1
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.

4 total

Σ	#	Presentation	Mapping
7	541	⟨a, b, c \| ba=ac, ca=b⟩	φ(a) = b, φ(b) = ab, φ(c) = a
7	1033	⟨a, b, c \| ab=c, baa=c⟩	φ(a) = b, φ(b) = a, φ(c) = ba
8	5716	⟨a, b, c \| aa=b, acc=ca⟩	φ(a) = a, φ(b) = aa, φ(c) = b
8	6057	⟨a, b, c \| ab=c, baa=ab⟩	φ(a) = b, φ(b) = a, φ(c) = ba