#2701 ⟨a, b | abbaa=abaab⟩

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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 ⋅ b²a² = aba²b and a ⋅ ba²b = aba²b, however b²a² ≠ ba²b
Enveloping group: ⟨a, b | aaba^-1a^-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 = abbaab
Reduction order:
- Left-to-right shortlex with b < a < c
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	cba² ⇒ abac	[6]
2.	ca²b ⇒ abac	[5]
3.	cacb ⇒ cbac	[9]
4.	aba²b ⇒ ab²a²	[1]
5.	abacb ⇒ ab²ac	[8]
6.	caca² ⇒ ca³c	[14]
7.	ab²a²b ⇒ c	[2]
8.	cbaca² ⇒ ab(ac)²	[11]
9.	ca⁴b ⇒ abaca²	[12]
10.	ca³cb ⇒ ab(ac)²	[13]
11.	ab²aca² ⇒ ab²a³c	[10]
12.	ab²a⁴b ⇒ ca²	[4]
13.	ab²a³cb ⇒ cac	[7]

# ab:abbaa=abaab bac abbaab=c morph:6/1
cbaa=abac
caab=abac
cacb=cbac
abaab=abbaa
abacb=abbac
cacaa=caaac
abbaab=c
cbacaa=abacac
caaaab=abacaa
caaacb=abacac
abbacaa=abbaaac
abbaaaab=caa
abbaaacb=cac