#5421 ⟨a, b | aba=bb, bab=aa⟩

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Properties
Elements
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 32 elements
Not cancellative, because multiplication by a⁴ is not injective:
- a⁴ ⋅ a⁶ = a⁴ and a⁴ ⋅ 1 = a⁴, however a⁶ ≠ 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.

4 element center:
- 1, a³, a⁶, a⁹
1 non-trivial idempotent:
- a⁶
Order of generators:
- a: index 4, period 6
- b: index 4, period 6

Histogram:

index 1, period 1	1 element	a⁶
index 1, period 2	1 element	a⁹
index 1, period 3	8 elements	a²b, ba², a⁴, a²ba, aba², ...
index 1, period 4	6 elements	a²ba², aba³, ba⁴, a²ba⁵, aba⁶, ...
index 1, period 6	8 elements	a⁵, a²ba³, aba⁴, ba⁵, a⁷, ...
index 2, period 2	1 element	a³
index 2, period 3	2 elements	a², aba
index 3, period 4	2 elements	ab, ba
index 4, period 6	2 elements	a, b

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.	a¹⁰ ⇒ a⁴	[8]
2.	ba⁸ ⇒ ba²	[7]
3.	a²ba⁶ ⇒ a²b	[9]
4.	a³b ⇒ ba³	[5]
5.	b² ⇒ aba	[1]
6.	bab ⇒ a²	[2]
7.	ba²b ⇒ a²ba⁵	[6]

# ab:aba=bb,bab=aa a/b
aaaaaaaaaa=aaaa
baaaaaaaa=baa
aabaaaaaa=aab
aaab=baaa
bb=aba
bab=aa
baab=aabaaaaa

Same cardinality

2 unique, 19 total

Σ	#	Presentation	Description	Related
9	3385	⟨a, b \| aa=1, abbbba=b⟩	Finite non-commutative monoid with 32 elements	17 iso
11	15515	⟨a, b \| aaa=bb, aabab=a⟩	Finite non-commutative monoid with 32 elements

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