#7050 ⟨a, b | bb=aa, ababa=a⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 26 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.

5 element center:
- 1, a², a⁴, a⁶, a⁸
3 non-trivial idempotents:
- (ab)², (ba)², a⁸
Order of generators:
- a: index 2, period 8
- b: index 2, period 8

Histogram:

index 1, period 1	3 elements	(ab)², (ba)², a⁸
index 1, period 2	5 elements	ab, ba, a⁴, ba⁵, ba⁹
index 1, period 4	4 elements	a², ba³, a⁶, ba⁷
index 1, period 8	8 elements	a³, ba², a⁵, ba⁴, a⁷, ...
index 2, period 8	5 elements	a, b, aba, bab, b(ab)²

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

a¹⁰ ⇒ a²
aba² ⇒ ba⁷
a²b ⇒ ba²
b² ⇒ a²
a(ba)² ⇒ a

# ab:bb=aa,ababa=a a/b
aaaaaaaaaa=aa
abaa=baaaaaaa
aab=baa
bb=aa
ababa=a

Right Cayley graph

Left Cayley graph

Others with same cardinality

5 unique, 26 total

Σ	#	Presentation	Description	Related
9	3645	⟨a, b \| aa=1, abab=bbb⟩	Finite non-commutative monoid with 26 elements	12 iso, 7 anti-iso
10	5088	⟨a, b \| aaa=bb, baab=a⟩	Finite non-commutative monoid with 26 elements	1 iso
11	15941	⟨a, b \| aba=bb, bbabb=a⟩	Finite non-commutative monoid with 26 elements
11	19766	⟨a, b \| aba=b, babbb=aa⟩	Finite non-commutative monoid with 26 elements
11	26639	⟨a, b \| aa=1, abbbbbbb=b⟩	Finite non-commutative monoid with 26 elements	1 iso

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