#5913 ⟨a, b | abbaab=aabaa⟩

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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 ⋅ b²a²b = a²ba² and a ⋅ aba² = a²ba², however b²a²b ≠ aba²
Enveloping group: ⟨a, b | aabab^-1aabb⟩

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 = aba
2. d = bba
Reduction order:
- Left-to-right shortlex with a < b < d < c
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	aba ⇒ c	[2]
2.	b²a ⇒ d	[3]
3.	dba ⇒ b²c	[7]
4.	cba ⇒ abc	[6]
5.	adab ⇒ aca	[5]
6.	(ad)² ⇒ ac²	[11]
7.	aca² ⇒ adc	[10]
8.	d²ab ⇒ dca	[9]
9.	d²ad ⇒ dc²	[13]
10.	dca² ⇒ d²c	[17]
11.	cdab ⇒ c²a	[8]
12.	cdad ⇒ c³	[12]
13.	c²a² ⇒ cdc	[16]
14.	ac²ab ⇒ adaca	[14]
15.	ac²ad ⇒ adac²	[15]
16.	dc²ab ⇒ d²aca	[18]
17.	dc²ad ⇒ d²ac²	[19]
18.	c³ab ⇒ cdaca	[20]
19.	c³ad ⇒ cdac²	[21]

# ab:abbaab=aabaa abdc aba=c,bba=d morph:3/2,3/0
aba=c
bba=d
dba=bbc
cba=abc
adab=aca
adad=acc
acaa=adc
ddab=dca
ddad=dcc
dcaa=ddc
cdab=cca
cdad=ccc
ccaa=cdc
accab=adaca
accad=adacc
dccab=ddaca
dccad=ddacc
cccab=cdaca
cccad=cdacc