#5659 ⟨a, b | aabaab=abbba⟩

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Properties
Rewriting system

Properties

Presentation has sum-of-sides 11
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ aba²b = ab³a and a ⋅ b³a = ab³a, however aba²b ≠ b³a
Enveloping group: ⟨a, b | aaabbaba^-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 = aab
Reduction order:
- Left-to-right recursive path with deg(c) = 0; deg(b) = deg(a) = 1, b < a
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	cb²a ⇒ ac²	[4]
2.	a²b ⇒ c	[2]
3.	ac²ab ⇒ cb²c	[7]
4.	c(b²c)² ⇒ ac⁴ab	[8]
5.	c²b³a ⇒ c²abc	[6]
6.	ab³c ⇒ c²ab	[5]
7.	ac²b³c ⇒ cb²c²ab	[10]
8.	c(cab)² ⇒ c²b³c	[9]
9.	c²b(b²c)² ⇒ c²abc³ab	[13]
10.	c²abcb³c ⇒ c²b³c²ab	[12]
11.	ab³a ⇒ c²	[3]
12.	c²abcb³a ⇒ c²b³c²	[11]

# ab:aabaab=abbba c/ba aab=c morph:3/0
cbba=acc
aab=c
accab=cbbc
cbbcbbc=accccab
ccbbba=ccabc
abbbc=ccab
accbbbc=cbbccab
ccabcab=ccbbbc
ccbbbcbbc=ccabcccab
ccabcbbbc=ccbbbccab
abbba=cc
ccabcbbba=ccbbbcc