#15941 ⟨a, b | aba=bb, bbabb=a⟩

Properties

Presentation has sum-of-sides 11
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, b³, b⁶, b⁹, b¹²
1 non-trivial idempotent:
- b¹²
Order of generators:
- a: index 1, period 8
- b: index 2, period 12

Histogram:

index 1, period 1	1 element	b¹²
index 1, period 2	1 element	b⁶
index 1, period 3	2 elements	b⁴, b⁸
index 1, period 4	2 elements	b³, b⁹
index 1, period 6	2 elements	b², b¹⁰
index 1, period 8	12 elements	a, ab, ab², ab³, ab⁴, ...
index 1, period 12	4 elements	b⁵, b⁷, b¹¹, b¹³
index 2, period 12	1 element	b

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

b¹⁴ ⇒ b²
ab¹² ⇒ a
ba ⇒ ab⁵
a² ⇒ b⁹

# ab:aba=bb,bbabb=a b/a
bbbbbbbbbbbbbb=bb
abbbbbbbbbbbb=a
ba=abbbbb
aa=bbbbbbbbb

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
10	7050	⟨a, b \| bb=aa, ababa=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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