#15504 ⟨a, b | aaa=ab, bbbbb=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 55 elements
Not cancellative, because multiplication by a³ is not injective:
- a³ ⋅ a⁸ = a³ and a³ ⋅ 1 = a³, 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.

1 element center:
- 1
6 non-trivial idempotents:
- b⁴, b²a⁴, ba⁶, a⁸, b⁴a⁸, b³a¹⁰
Order of generators:
- a: index 3, period 8
- b: index 1, period 4

Histogram:

index 1, period 1	6 elements	b⁴, b²a⁴, ba⁶, a⁸, b⁴a⁸, ...
index 1, period 2	6 elements	b², a⁴, b⁴a⁴, b³a⁶, b²a⁸, ...
index 1, period 4	12 elements	b, b³, ba⁴, a⁶, b³a⁴, ...
index 1, period 8	20 elements	a³, ba³, a⁵, b²a³, ba⁵, ...
index 2, period 1	1 element	b³a²
index 2, period 2	1 element	ba²
index 2, period 4	3 elements	a², b²a², b⁴a²
index 2, period 8	4 elements	ba, b²a, b³a, b⁴a
index 3, period 8	1 element	a

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.	a¹¹ ⇒ a³	[3]
2.	ab ⇒ a³	[1]
3.	b⁵ ⇒ b	[2]

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

Same cardinality

1 unique, 6 total

Σ	#	Presentation	Description	Related
10	4413	⟨a, b \| aaab=a, bbbbb=1⟩	Finite non-commutative monoid with 55 elements	5 iso

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