⟨a, b | aabbaa=aaba⟩

Up:

Monoids with two generators and one relation

#2522 ⟨a, b | aabbaa=aaab⟩

#2524 ⟨a, b | aabbaa=aabb⟩

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 ⋅ ab²a² = a²ba and a ⋅ aba = a²ba, however ab²a² ≠ aba

Not right cancellative, because right multiplication by a is not injective:

a²b²a ⋅ a = a²ba and a²b ⋅ a = a²ba, however a²b²a ≠ a²b

Enveloping group is isomorphic to ℤ

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 = aaba
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.

#	Rule	Proof
1.	a²ba ⇒ c	[2]
2.	caba ⇒ a²bc	[4]
3.	cb²a² ⇒ cba	[7]
4.	cb²ac ⇒ cbc	[11]
5.	c(ba)² ⇒ cb²c	[12]
6.	a²b²c ⇒ cba	[14]
7.	a²b²a² ⇒ c	[3]
8.	a²b²ac ⇒ a²bc	[8]
9.	cab²c ⇒ a²bcba	[15]
10.	cab²a² ⇒ a²bc	[17]
11.	cab²ac ⇒ cabc	[9]
12.	cbab²c ⇒ cb²cba	[16]
13.	cbab²a² ⇒ cb²c	[10]
14.	cbab²ac ⇒ cbabc	[13]

# ab:aabbaa=aaba ac/b aaba=c magic:0 aaba=c caba=aabc cbbaa=cba cbbac=cbc cbaba=cbbc aabbc=cba aabbaa=c aabbac=aabc cabbc=aabcba cabbaa=aabc cabbac=cabc cbabbc=cbbcba cbabbaa=cbbc cbabbac=cbabc

#2523 ⟨a, b | aabbaa=aaba⟩

Quick links

Properties

Rewriting system

Up:	Monoids with two generators and one relation
Prev:	#2522 ⟨a, b \| aabbaa=aaab⟩
Next:	#2524 ⟨a, b \| aabbaa=aabb⟩