#575 ⟨a, b | abba=abab⟩

Quick links

Properties
Rewriting system

Properties

Presentation has sum-of-sides 8
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ b²a = (ab)² and a ⋅ bab = (ab)², however b²a ≠ bab
Enveloping group: ⟨a, b | aba^-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 = abbab
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.	cba ⇒ abc	[6]
2.	cab ⇒ abc	[5]
3.	c²b ⇒ cbc	[9]
4.	c²a ⇒ cac	[14]
5.	(ab)² ⇒ ab²a	[1]
6.	abcb ⇒ ab²c	[8]
7.	cbca ⇒ abc²	[11]
8.	ca²b ⇒ abca	[12]
9.	cacb ⇒ abc²	[13]
10.	ab²ab ⇒ c	[2]
11.	ab²ca ⇒ ab²ac	[10]
12.	ab²a²b ⇒ ca	[4]
13.	ab²acb ⇒ c²	[7]

# ab:abba=abab bac abbab=c morph:5/0
cba=abc
cab=abc
ccb=cbc
cca=cac
abab=abba
abcb=abbc
cbca=abcc
caab=abca
cacb=abcc
abbab=c
abbca=abbac
abbaab=ca
abbacb=cc