#14861 ⟨a, b | abba=b, baaab=a⟩

Quick links

Properties
Elements
Right Cayley graph
Left Cayley graph
Rewriting system

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 49 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a⁸ = a and a ⋅ 1 = a, however a⁸ ≠ 1

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x² = x.
The index and period of x is the least m (index) and n (period) such that x^(m+n) = x^m.

3 element center:
- 1, a⁴, a⁸
1 non-trivial idempotent:
- a⁸
Order of generators:
- a: index 1, period 8
- b: index 1, period 6

Histogram:

index 1, period 1	1 element	a⁸
index 1, period 2	1 element	a⁴
index 1, period 3	8 elements	a³ba, a²ba², aba³, ba⁴, a³ba³, ...
index 1, period 4	18 elements	a², bab, a³b, a²ba, aba², ...
index 1, period 6	8 elements	b, a²b, aba, ba², a³ba⁵, ...
index 1, period 8	12 elements	a, ab, ba, a³, a⁵, ...

Right Cayley graph

Left Cayley graph

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

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

#	Rule	Proof
1.	a⁹ ⇒ a	[12]
2.	ba⁸ ⇒ b	[11]
3.	a⁴b ⇒ ba⁴	[4]
4.	b² ⇒ a³ba³	[8]
5.	ba²b ⇒ aba	[3]
6.	ba³b ⇒ a	[2]
7.	(ab)² ⇒ baba⁷	[9]

# ab:abba=b,baaab=a a/b
aaaaaaaaa=a
baaaaaaaa=b
aaaab=baaaa
bb=aaabaaa
baab=aba
baaab=a
abab=babaaaaaaa

Up:	Monoids with two generators and two relations
Prev:	#14843 ⟨a, b \| abba=b, aabab=a⟩
Next:	#14869 ⟨a, b \| abba=b, bbabb=a⟩