#2005 ⟨a, b | aabbbaba=ab⟩

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Properties
Rewriting system

Properties

Presentation has sum-of-sides 10
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ ab³aba = ab and a ⋅ b = ab, however ab³aba ≠ b
Enveloping group: ⟨a, b | aabbab^-1b^-1⟩

Rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

Auxiliary generators:
1. c = abbb
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.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	acaba ⇒ ab	[3]
2.	acabc ⇒ cb	[4]
3.	c²aba ⇒ cb	[10]
4.	abcaba ⇒ ab²	[5]
5.	cb² ⇒ (abc)²	[6]
6.	cbcaba ⇒ (abc)²	[11]
7.	acab² ⇒ c(ba)²	[8]
8.	c²abcb ⇒ cbcabc	[13]
9.	ab³ ⇒ c	[2]
10.	ab²caba ⇒ c	[7]
11.	(abc)²b ⇒ ab²cabc	[9]
12.	cb(ab)² ⇒ ac²	[14]
13.	cbcabcb ⇒ ab(cabc)²	[15]
14.	ab²cabcb ⇒ c²abc	[12]

# ab:aabbbaba=ab ac/b abbb=c morph:4/0
acaba=ab
acabc=cb
ccaba=cb
abcaba=abb
cbb=abcabc
cbcaba=abcabc
acabb=cbaba
ccabcb=cbcabc
abbb=c
abbcaba=c
abcabcb=abbcabc
cbabab=acc
cbcabcb=abcabccabc
abbcabcb=ccabc