#3439 ⟨a, b, c | ba=ac, aab=c⟩

Properties

Sum of relation sides is 8
Infinite cancellative non-commutative monoid
Element a has infinite order
Submonoid of enveloping group: ⟨a, b | aaaba^-1b^-1⟩
- Embedding: φ(a) = a, φ(b) = aba^-1, 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(b) = 0, a < b; deg(c) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a³b ⇒ ba	[3]
2.	c ⇒ a²b	[2]

# abc:ba=ac,aab=c ab/c - -
aaab=ba
c=aab

5 unique, 11 total

Σ	#	Presentation	Properties	φ
8	1997	⟨a, b, c \| abc=bb, cba=1⟩	Grp Inf	5
8	3465	⟨a, b, c \| ba=ac, bbb=c⟩	Can Inf	1
8	3540	⟨a, b, c \| bb=ac, cba=b⟩	Can Inf
8	5592	⟨a, b, c \| ab=c, bcca=c⟩	Can Inf
8	6110	⟨a, b, c \| ab=c, ccc=ba⟩	Can Inf

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

3 total

Σ	#	Presentation	Mapping
8	5692	⟨a, b, c \| aa=b, abc=ca⟩	φ(a) = a, φ(b) = aa, φ(c) = b
8	5728	⟨a, b, c \| aa=b, bac=ca⟩	φ(a) = a, φ(b) = aa, φ(c) = b
8	6008	⟨a, b, c \| ab=c, aac=ba⟩	φ(a) = a, φ(b) = b, φ(c) = ab