#19302 ⟨a, b | aaa=a, bbbbb=ab⟩

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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 39 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.

1 element center:
- 1
4 non-trivial idempotents:
- a², b⁸, b⁸a², b¹²a
Order of generators:
- a: index 1, period 2
- b: index 5, period 8

Histogram:

index 1, period 1	4 elements	a², b⁸, b⁸a², b¹²a
index 1, period 2	4 elements	a, b⁸a, b¹², b¹²a²
index 1, period 4	6 elements	b⁶, b⁶a, b⁶a², b¹⁰, b¹⁰a, ...
index 1, period 8	12 elements	b⁵, b⁵a, b⁵a², b⁷, b⁷a, ...
index 2, period 1	1 element	b⁴a
index 2, period 2	2 elements	b⁴, b⁴a²
index 2, period 4	2 elements	b²a, b²a²
index 2, period 8	5 elements	ba, ba², b³, b³a, b³a²
index 3, period 4	1 element	b²
index 5, period 8	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(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b¹³ ⇒ b⁵	[3]
2.	ab ⇒ b⁵	[2]
3.	a³ ⇒ a	[1]

# ab:aaa=a,bbbbb=ab b/a
bbbbbbbbbbbbb=bbbbb
ab=bbbbb
aaa=a

Same cardinality

1 unique, 5 total

Σ	#	Presentation	Description	Related
10	7531	⟨a, b \| aaa=1, abbbbb=b⟩	Finite non-commutative monoid with 39 elements	4 iso

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