#11163 ⟨a, b | aaaab=a, bbbbb=1⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 80 elements

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x² = x.
The order of x is the least n (if it exists) such that xⁿ = 1.
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:
- ba³, b²a⁶, b³a⁹, a¹⁵, b⁴a¹²
Order of generators:
- a: index 1, period 15
- b: order 5

Histogram:

order 5	4 elements	b, b², b³, b⁴
index 1, period 1	5 elements	ba³, b²a⁶, b³a⁹, a¹⁵, b⁴a¹²
index 1, period 3	10 elements	b²a, a⁵, b⁴a², b³a⁴, ba⁸, ...
index 1, period 5	20 elements	a³, b²a³, a⁶, b³a³, ba⁶, ...
index 1, period 15	40 elements	a, a², ba, ba², a⁴, ...

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

a¹⁶ ⇒ a
ab ⇒ a¹³
b⁵ ⇒ 1

# ab:aaaab=a,bbbbb=1 a/b
aaaaaaaaaaaaaaaa=a
ab=aaaaaaaaaaaaa
bbbbb=1

Right Cayley graph

Left Cayley graph

Other isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

1 total

Σ	#	Presentation	Mapping
11	11671	⟨a, b \| aaaa=ab, bbbbb=1⟩	φ(a) = baa, φ(b) = b

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