#2082 ⟨a, b | abbabbba=ab⟩

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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 ⋅ b²ab³a = ab and a ⋅ b = ab, however b²ab³a ≠ b
Enveloping group: ⟨a, b | aabb⟩

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

#	Rule	Proof
1.	c³a ⇒ cac²	[30]
2.	c(ca)² ⇒ c(ac)²	[26]
3.	c²a²ca ⇒ caca²c	[27]
4.	(c²a)² ⇒ c²a²c²	[31]
5.	(ca)²c²a ⇒ caca²c²	[29]
6.	c²a²c²a ⇒ c²a³c²	[32]
7.	(ca)⁴ ⇒ c(aca)²c	[15]
8.	c²a³cac ⇒ caca²c²a	[28]
9.	(ca)³(ac)² ⇒ c(aca)²c²a	[23]
10.	caca²(ca)³ ⇒ c	[17]
11.	cac(a(ac)²)² ⇒ c²a	[21]
12.	ab ⇒ c(aca²c)²a²	[19]
13.	cb ⇒ c²a	[24]

# ab:abbabbba=ab ac/b abbbbb=c morph:6/0
ccca=cacc
ccaca=cacac
ccaaca=cacaac
ccacca=ccaacc
cacacca=cacaacc
ccaacca=ccaaacc
cacacaca=cacaacac
ccaaacac=cacaacca
cacacaacac=cacaacacca
cacaacacaca=c
cacaacacaacac=cca
ab=cacaacacaacaa
cb=cca