#15515 ⟨a, b | aaa=bb, aabab=a⟩

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

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 32 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a¹⁵ = a and a ⋅ 1 = a, however a¹⁵ ≠ 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.

6 element center:
- 1, a³, a⁶, a⁹, a¹², a¹⁵
1 non-trivial idempotent:
- a¹⁵
Order of generators:
- a: index 1, period 15
- b: index 2, period 10

Histogram:

index 1, period 1	1 element	a¹⁵
index 1, period 2	3 elements	ba, ba⁶, ba¹¹
index 1, period 3	2 elements	a⁵, a¹⁰
index 1, period 5	4 elements	a³, a⁶, a⁹, a¹²
index 1, period 10	12 elements	ba², ba³, ba⁴, ba⁵, ba⁷, ...
index 1, period 15	8 elements	a, a², a⁴, a⁷, a⁸, ...
index 2, period 10	1 element	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.	a¹⁶ ⇒ a	[8]
2.	ab ⇒ ba¹¹	[7]
3.	b² ⇒ a³	[1]

# ab:aaa=bb,aabab=a a/b
aaaaaaaaaaaaaaaa=a
ab=baaaaaaaaaaa
bb=aaa

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
10	5421	⟨a, b \| aba=bb, bab=aa⟩	Finite non-commutative monoid with 32 elements

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