#3431 ⟨a, b, c | ba=ab, cac=b⟩

Properties

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

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

Auxiliary generators:
1. d = ac
Reduction order:
- Left-to-right recursive path with deg(d) = 0; deg(c) = deg(b) = deg(a) = 1, c < b < a
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

# abc:ba=ab,cac=b d/cba ac=d morph:2/0
ddc=cdd
b=cd
dda=add
cda=dd
ac=d

8 unique, 16 total

Σ	#	Presentation	Properties	φ
7	533	⟨a, b, c \| ba=ab, cc=a⟩	Can Inf
8	2199	⟨a, b, c \| aabc=1, acba=1⟩	Grp Inf	5
8	2976	⟨a, b, c \| aab=c, baa=c⟩	Can Inf	2
8	3072	⟨a, b, c \| abc=b, cba=b⟩	Can Inf
8	3587	⟨a, b, c \| ba=ab, cc=ab⟩	Can Inf	1
8	3592	⟨a, b, c \| ba=ac, ca=ab⟩	Can Inf
8	3596	⟨a, b, c \| ba=ac, cc=bb⟩	Can Inf
8	6088	⟨a, b, c \| ab=c, bca=cc⟩	Can Inf