#19756 ⟨a, b | aba=b, baaab=aa⟩

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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 a² is not injective:
- a² ⋅ a¹⁰ = a² and a² ⋅ 1 = a², however a¹⁰ ≠ 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, a⁵, a¹⁰
1 non-trivial idempotent:
- a¹⁰
Order of generators:
- a: index 2, period 10
- b: index 1, period 4

Histogram:

index 1, period 1	1 element	a¹⁰
index 1, period 2	1 element	a⁵
index 1, period 4	10 elements	b, ba, ba², ba³, ba⁴, ...
index 1, period 5	4 elements	a², a⁴, a⁶, a⁸
index 1, period 10	4 elements	a³, a⁷, a⁹, a¹¹
index 2, period 10	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	ba⁸	a¹⁰	ba⁹	a¹¹
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹
a	a	a²	ba⁹	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a¹¹	ba⁸	a²
b	b	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a¹¹	ba⁸	a²	ba⁹	a³	b	a⁴	ba
a²	a²	a³	ba⁸	a⁴	ba⁹	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a¹¹	ba⁶	a²	ba⁷	a³
ba	ba	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	b	a²	ba	a³	ba²
a³	a³	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba⁹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	a²	ba⁵	a³	ba⁶	a⁴
ba²	ba²	ba³	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba⁹	a⁹	b	a¹⁰	ba	a¹¹	ba²	a²	ba³
a⁴	a⁴	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba⁹	a⁹	b	a¹⁰	ba	a¹¹	ba²	a²	ba³	a³	ba⁴	a⁴	ba⁵	a⁵
ba³	ba³	ba⁴	a²	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba⁹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴
a⁵	a⁵	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶
ba⁴	ba⁴	ba⁵	a¹¹	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵
a⁶	a⁶	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a¹¹	ba⁸	a²	ba⁹	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷
ba⁵	ba⁵	ba⁶	a¹⁰	ba⁷	a¹¹	ba⁸	a²	ba⁹	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶
a⁷	a⁷	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a¹¹	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	b	a⁶	ba	a⁷	ba²	a⁸
ba⁶	ba⁶	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷
a⁸	a⁸	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	a²	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba⁹	a⁷	b	a⁸	ba	a⁹
ba⁷	ba⁷	ba⁸	a⁸	ba⁹	a⁹	b	a¹⁰	ba	a¹¹	ba²	a²	ba³	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸
a⁹	a⁹	a¹⁰	ba	a¹¹	ba²	a²	ba³	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba⁹	a⁹	b	a¹⁰
ba⁸	ba⁸	ba⁹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	a²	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba⁹
a¹⁰	a¹⁰	a¹¹	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹
ba⁹	ba⁹	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a¹¹	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	b
a¹¹	a¹¹	a²	ba⁹	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a¹¹	ba⁸	a²

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 ⇒ a²	[9]
2.	ba¹⁰ ⇒ b	[8]
3.	ab ⇒ ba⁹	[7]
4.	b² ⇒ a⁵	[5]

# ab:aba=b,baaab=aa a/b
aaaaaaaaaaaa=aa
baaaaaaaaaa=b
ab=baaaaaaaaa
bb=aaaaa

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	24722	⟨a, b \| aa=a, bbbbbb=ab⟩	Finite non-commutative monoid with 22 elements

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