#12195 ⟨a, b | aaaa=ab, bbbb=b⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 52 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:
- b³, ba⁶, a⁹, b³a⁹, b²a¹²
Order of generators:
- a: index 4, period 9
- b: index 1, period 3

Histogram:

index 1, period 1	5 elements	b³, ba⁶, a⁹, b³a⁹, b²a¹²
index 1, period 3	10 elements	b, b², a⁶, b²a⁶, b³a⁶, ...
index 1, period 9	24 elements	a⁴, a⁵, ba⁴, ba⁵, b²a⁴, ...
index 2, period 1	1 element	b²a³
index 2, period 3	3 elements	a³, ba³, b³a³
index 2, period 9	7 elements	a², ba, ba², b²a, b²a², ...
index 4, period 9	1 element	a

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(a) = 0; deg(b) = 1

a¹³ ⇒ a⁴
ab ⇒ a⁴
b⁴ ⇒ b

# ab:aaaa=ab,bbbb=b a/b
aaaaaaaaaaaaa=aaaa
ab=aaaa
bbbb=b

Right Cayley graph

Left Cayley graph

Others with same cardinality

1 unique, 4 total

Σ	#	Presentation	Description	Related
10	3767	⟨a, b \| aaaa=ab, bbbb=1⟩	Finite non-commutative monoid with 52 elements	3 anti-iso

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