#6764 ⟨a, b | aba=b, aaaa=bb⟩

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

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 20 elements
Not cancellative, because multiplication by a⁴ is not injective:
- a⁴ ⋅ a⁸ = a⁴ and a⁴ ⋅ 1 = a⁴, however a⁸ ≠ 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.

3 element center:
- 1, a⁴, a⁸
1 non-trivial idempotent:
- a⁸
Order of generators:
- a: index 4, period 8
- b: index 1, period 4

Histogram:

index 1, period 1	1 element	a⁸
index 1, period 2	1 element	a⁴
index 1, period 4	10 elements	b, ba, ba², ba³, ba⁴, ...
index 1, period 8	4 elements	a⁵, a⁷, a⁹, a¹¹
index 2, period 4	1 element	a²
index 2, period 8	1 element	a³
index 4, period 8	1 element	a

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	a¹⁰	a¹¹
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	a¹⁰	a¹¹
a	a	a²	ba⁷	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	a¹¹	a⁴
b	b	ba	a⁴	ba²	a⁵	ba³	a⁶	ba⁴	a⁷	ba⁵	a⁸	ba⁶	a⁹	ba⁷	a¹⁰	b	a¹¹	ba	ba²	ba³
a²	a²	a³	ba⁶	a⁴	ba⁷	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a¹¹	a⁴	a⁵
ba	ba	ba²	a¹¹	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	b	a⁹	ba	a¹⁰	ba²	ba³	ba⁴
a³	a³	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	a⁴	a⁵	a⁶
ba²	ba²	ba³	a¹⁰	ba⁴	a¹¹	ba⁵	a⁴	ba⁶	a⁵	ba⁷	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	ba⁴	ba⁵
a⁴	a⁴	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	b	a¹⁰	ba	a¹¹	ba²	a⁴	ba³	a⁵	a⁶	a⁷
ba³	ba³	ba⁴	a⁹	ba⁵	a¹⁰	ba⁶	a¹¹	ba⁷	a⁴	b	a⁵	ba	a⁶	ba²	a⁷	ba³	a⁸	ba⁴	ba⁵	ba⁶
a⁵	a⁵	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a¹¹	b	a⁴	ba	a⁵	ba²	a⁶	a⁷	a⁸
ba⁴	ba⁴	ba⁵	a⁸	ba⁶	a⁹	ba⁷	a¹⁰	b	a¹¹	ba	a⁴	ba²	a⁵	ba³	a⁶	ba⁴	a⁷	ba⁵	ba⁶	ba⁷
a⁶	a⁶	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a¹¹	ba⁶	a⁴	ba⁷	a⁵	b	a⁶	ba	a⁷	a⁸	a⁹
ba⁵	ba⁵	ba⁶	a⁷	ba⁷	a⁸	b	a⁹	ba	a¹⁰	ba²	a¹¹	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	ba⁷	b
a⁷	a⁷	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	b	a⁸	a⁹	a¹⁰
ba⁶	ba⁶	ba⁷	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	a¹⁰	ba⁴	a¹¹	ba⁵	a⁴	ba⁶	a⁵	ba⁷	b	ba
a⁸	a⁸	a⁹	b	a¹⁰	ba	a¹¹	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	a¹⁰	a¹¹
ba⁷	ba⁷	b	a⁵	ba	a⁶	ba²	a⁷	ba³	a⁸	ba⁴	a⁹	ba⁵	a¹⁰	ba⁶	a¹¹	ba⁷	a⁴	b	ba	ba²
a⁹	a⁹	a¹⁰	ba⁷	a¹¹	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	a¹¹	a⁴
a¹⁰	a¹⁰	a¹¹	ba⁶	a⁴	ba⁷	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a¹¹	a⁴	a⁵
a¹¹	a¹¹	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	a⁴	a⁵	a⁶

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(a) = 0; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a¹² ⇒ a⁴	[8]
2.	ba⁸ ⇒ b	[7]
3.	ab ⇒ ba⁷	[6]
4.	b² ⇒ a⁴	[2]

# ab:aba=b,aaaa=bb a/b
aaaaaaaaaaaa=aaaa
baaaaaaaa=b
ab=baaaaaaa
bb=aaaa

Same cardinality

20 unique, 65 total

Σ	#	Presentation	Description	Related
8	526	⟨a, b \| aab=a, bbbb=1⟩	Finite non-commutative monoid with 20 elements	10 iso, 9 anti-iso
9	2207	⟨a, b \| ab=aa, bbbb=b⟩	Finite non-commutative monoid with 20 elements	1 anti-iso
10	4095	⟨a, b \| baa=abb, abab=1⟩	Finite non-Abelian group with 20 elements	5 iso
10	4212	⟨a, b \| aaaaa=1, abbbb=1⟩	Isomorphic to ℤ₂₀	16 iso
10	5349	⟨a, b \| aaa=bb, bab=aa⟩	Finite non-commutative monoid with 20 elements	1 iso
10	6728	⟨a, b \| aba=a, aaaa=bb⟩	Finite non-commutative monoid with 20 elements
10	7117	⟨a, b \| ab=aa, bbbb=bb⟩	Finite non-commutative monoid with 20 elements
11	12159	⟨a, b \| aaaa=aa, abbb=b⟩	Finite non-commutative monoid with 20 elements
11	12322	⟨a, b \| aaab=bb, bbba=a⟩	Finite non-commutative monoid with 20 elements
11	14367	⟨a, b \| aaaa=a, bbbbb=a⟩	Isomorphic to ℕ_{(20 = 5)}
11	14407	⟨a, b \| aaaa=b, bbbbb=a⟩	Isomorphic to ℕ_{(20 = 1)}
11	14408	⟨a, b \| aaaa=b, bbbbb=b⟩	Isomorphic to ℕ_{(20 = 4)}
11	15933	⟨a, b \| aba=bb, baaab=a⟩	Finite non-commutative monoid with 20 elements
11	16124	⟨a, b \| aab=aa, baba=bb⟩	Finite non-commutative monoid with 20 elements
11	16339	⟨a, b \| aba=aa, aaaa=bb⟩	Finite non-commutative monoid with 20 elements
11	16343	⟨a, b \| aba=aa, aaab=bb⟩	Finite non-commutative monoid with 20 elements
11	16545	⟨a, b \| aba=bb, abb=aaa⟩	Finite non-commutative monoid with 20 elements
11	18619	⟨a, b \| aba=b, aaaaabb=1⟩	Finite non-Abelian group with 20 elements	3 iso
11	19503	⟨a, b \| aab=a, bbbbb=bb⟩	Finite non-commutative monoid with 20 elements
11	21047	⟨a, b \| ab=aa, bbbb=bbb⟩	Finite non-commutative monoid with 20 elements

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