#18737 ⟨a, b | aaa=a, abbbba=b⟩

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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 33 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a² = a and a ⋅ 1 = a, however a² ≠ 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.

5 element center:
- 1, a², b⁵, b¹⁰, b¹⁵
2 non-trivial idempotents:
- a², b¹⁵
Order of generators:
- a: index 1, period 2
- b: index 1, period 15

Histogram:

index 1, period 1	2 elements	a², b¹⁵
index 1, period 2	6 elements	a, ab³, ab⁶, ab⁹, ab¹², ...
index 1, period 3	2 elements	b⁵, b¹⁰
index 1, period 5	4 elements	b³, b⁶, b⁹, b¹²
index 1, period 6	10 elements	ab, ab², ab⁴, ab⁵, ab⁷, ...
index 1, period 15	8 elements	b, b², b⁴, b⁷, b⁸, ...

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(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b¹⁶ ⇒ b	[8]
2.	ba ⇒ ab⁴	[6]
3.	a²b ⇒ b	[3]
4.	a³ ⇒ a	[1]

# ab:aaa=a,abbbba=b b/a
bbbbbbbbbbbbbbbb=b
ba=abbbb
aab=b
aaa=a

Same cardinality

1 unique, 2 total

Σ	#	Presentation	Description	Related
11	15526	⟨a, b \| aaa=bb, ababb=b⟩	Finite non-commutative monoid with 33 elements	1 iso

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