#24722 ⟨a, b | aa=a, bbbbbb=ab⟩

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Properties
Elements
Cayley table
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 22 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a = a and a ⋅ 1 = a, however a ≠ 1

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
3 non-trivial idempotents:
- a, b¹⁰, b¹⁰a
Order of generators:
- a: index 1, period 1
- b: index 6, period 5

Histogram:

index 1, period 1	3 elements	a, b¹⁰, b¹⁰a
index 1, period 5	8 elements	b⁶, b⁶a, b⁷, b⁷a, b⁸, ...
index 2, period 1	2 elements	b⁵, b⁵a
index 2, period 5	6 elements	ba, b²a, b³, b³a, b⁴, ...
index 3, period 5	1 element	b²
index 6, period 5	1 element	b

Cayley table

Idempotents are shown in bold.

	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
1	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
a	a	a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
b	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a
ba	ba	ba	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a
b²	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a
b²a	b²a	b²a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a
b³	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b³a	b³a	b³a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁴	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a
b⁴a	b⁴a	b⁴a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a
b⁵	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
b⁵a	b⁵a	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
b⁶	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a
b⁶a	b⁶a	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a
b⁷	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a
b⁷a	b⁷a	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a
b⁸	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁸a	b⁸a	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁹	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a
b⁹a	b⁹a	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a
b¹⁰	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
b¹⁰a	b¹⁰a	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b¹¹ ⇒ b⁶	[3]
2.	ab ⇒ b⁶	[2]
3.	a² ⇒ a	[1]

# ab:aa=a,bbbbbb=ab b/a
bbbbbbbbbbb=bbbbbb
ab=bbbbbb
aa=a

Same cardinality

3 unique, 35 total

Σ	#	Presentation	Description	Related
9	3375	⟨a, b \| aa=1, ababba=b⟩	Finite non-commutative monoid with 22 elements	15 iso, 12 anti-iso
10	9759	⟨a, b \| aa=1, abbbbbb=b⟩	Finite non-commutative monoid with 22 elements	5 iso
11	19756	⟨a, b \| aba=b, baaab=aa⟩	Finite non-commutative monoid with 22 elements

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