#14356 ⟨a, b | aaaa=a, abbbb=b⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 40 elements

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
5 non-trivial idempotents:
- a³, b³a, b⁶a², b⁹, b⁹a³
Order of generators:
- a: index 1, period 3
- b: index 1, period 9
Histogram:

index 1, period 1 5 elements a³, b³a, b⁶a², b⁹, b⁹a³

index 1, period 3 10 elements a, a², b³, b³a², b³a³, ...

index 1, period 9 24 elements b, ba, b², ba², b²a, ...

index 1, period 1	5 elements	a³, b³a, b⁶a², b⁹, b⁹a³
index 1, period 3	10 elements	a, a², b³, b³a², b³a³, ...
index 1, period 9	24 elements	b, ba, b², ba², b²a, ...

Complete 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

b¹⁰ ⇒ b
ab ⇒ b⁷
a⁴ ⇒ a

# ab:aaaa=a,abbbb=b b/a
bbbbbbbbbb=b
ab=bbbbbbb
aaaa=a

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#14356 ⟨a, b | aaaa=a, abbbb=b⟩

Properties

Elements

Complete rewriting system

Right Cayley graph

Left Cayley graph