#2552 ⟨a, b | aabbba=abab⟩

Quick links

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 ⋅ ab³a = (ab)² and a ⋅ bab = (ab)², however ab³a ≠ bab
Enveloping group: ⟨a, b | aaabb⟩

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 = abab
Reduction order:
- Left-to-right recursive path with deg(a) = 0; deg(b) = deg(c) = 1, b < c
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	(ab)² ⇒ c	[2]
2.	cab ⇒ abc	[4]
3.	a²b³a ⇒ c	[3]
4.	a²b³c ⇒ cbab	[6]
5.	abcb²a ⇒ cbab	[8]
6.	c²b²a ⇒ abcbab	[10]
7.	c(bab)² ⇒ abcb²c	[7]
8.	cbcb²a ⇒ abcb²c	[9]
9.	a(bcb)²ab ⇒ cbcb²c	[12]
10.	cbab²cb²a ⇒ cbcb²c	[16]
11.	c²b²cbab ⇒ a(bc)²b²c	[13]
12.	c(bcb)²ab ⇒ cba(b²c)²	[14]
13.	cba(b²c)²bab ⇒ a(bcb²c)²	[15]

# ab:aabbba=abab a/bc abab=c magic:0
abab=c
cab=abc
aabbba=c
aabbbc=cbab
abcbba=cbab
ccbba=abcbab
cbabbab=abcbbc
cbcbba=abcbbc
abcbbcbab=cbcbbc
cbabbcbba=cbcbbc
ccbbcbab=abcbcbbc
cbcbbcbab=cbabbcbbc
cbabbcbbcbab=abcbbcbcbbc