#13248 ⟨a, b | bab=aaa, bbb=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 21 elements
Not cancellative, because multiplication by a⁵ is not injective:
- a⁵ ⋅ a² = a⁵ and a⁵ ⋅ 1 = a⁵, however a² ≠ 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.

4 element center:
- 1, b³, b⁶, b⁹
1 non-trivial idempotent:
- b⁹
Order of generators:
- a: index 5, period 2
- b: index 7, period 3

Histogram:

index 1, period 1	1 element	b⁹
index 1, period 2	3 elements	ab⁴, ab⁵, ab⁶
index 1, period 3	2 elements	b⁷, b⁸
index 2, period 1	1 element	b⁶
index 2, period 2	3 elements	ab², b²a, ab³
index 2, period 3	4 elements	aba, ab²a, b⁴, b⁵
index 3, period 1	1 element	b³
index 3, period 2	2 elements	ab, ba
index 4, period 3	1 element	b²
index 5, period 2	1 element	a
index 7, period 3	1 element	b

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	b⁸	b⁹
1	1	a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	b⁸	b⁹
a	a	b³	ab	b⁴	aba	ab²	ab⁵	b⁵	ab²a	ab³	ab⁴	b⁶	ab⁴	b⁷	ab⁵	b⁸	ab⁶	b⁹	ab⁴	ab⁵	ab⁶
b	b	ba	b²	ab³	b²a	b³	b⁶	ab⁴	ab³	b⁴	b⁸	ab⁵	b⁵	ab⁶	b⁶	ab⁴	b⁷	ab⁵	b⁸	b⁹	b⁷
ab	ab	aba	ab²	b⁶	ab²a	ab³	ab⁶	b⁷	b⁶	ab⁴	ab⁵	b⁸	ab⁵	b⁹	ab⁶	b⁷	ab⁴	b⁸	ab⁵	ab⁶	ab⁴
ba	ba	b⁴	ab³	b⁵	b⁶	ab⁴	ab⁴	b⁶	b⁸	ab⁵	ab⁶	b⁷	ab⁶	b⁸	ab⁴	b⁹	ab⁵	b⁷	ab⁶	ab⁴	ab⁵
b²	b²	b²a	b³	ab⁵	ab³	b⁴	b⁷	ab⁶	ab⁵	b⁵	b⁹	ab⁴	b⁶	ab⁵	b⁷	ab⁶	b⁸	ab⁴	b⁹	b⁷	b⁸
aba	aba	ab⁴	b⁶	ab⁵	ab⁶	b⁷	b⁷	ab⁶	ab⁵	b⁸	b⁹	ab⁴	b⁹	ab⁵	b⁷	ab⁶	b⁸	ab⁴	b⁹	b⁷	b⁸
ab²	ab²	ab²a	ab³	b⁸	b⁶	ab⁴	ab⁴	b⁹	b⁸	ab⁵	ab⁶	b⁷	ab⁶	b⁸	ab⁴	b⁹	ab⁵	b⁷	ab⁶	ab⁴	ab⁵
b²a	b²a	b⁵	ab⁵	b⁶	b⁷	ab⁶	ab⁶	b⁷	b⁹	ab⁴	ab⁵	b⁸	ab⁵	b⁹	ab⁶	b⁷	ab⁴	b⁸	ab⁵	ab⁶	ab⁴
b³	b³	ab³	b⁴	ab⁴	ab⁵	b⁵	b⁸	ab⁵	ab⁴	b⁶	b⁷	ab⁶	b⁷	ab⁴	b⁸	ab⁵	b⁹	ab⁶	b⁷	b⁸	b⁹
ab²a	ab²a	ab⁵	b⁸	ab⁶	ab⁴	b⁹	b⁹	ab⁴	ab⁶	b⁷	b⁸	ab⁵	b⁸	ab⁶	b⁹	ab⁴	b⁷	ab⁵	b⁸	b⁹	b⁷
ab³	ab³	b⁶	ab⁴	b⁷	b⁸	ab⁵	ab⁵	b⁸	b⁷	ab⁶	ab⁴	b⁹	ab⁴	b⁷	ab⁵	b⁸	ab⁶	b⁹	ab⁴	ab⁵	ab⁶
b⁴	b⁴	ab⁵	b⁵	ab⁶	ab⁴	b⁶	b⁹	ab⁴	ab⁶	b⁷	b⁸	ab⁵	b⁸	ab⁶	b⁹	ab⁴	b⁷	ab⁵	b⁸	b⁹	b⁷
ab⁴	ab⁴	b⁸	ab⁵	b⁹	b⁷	ab⁶	ab⁶	b⁷	b⁹	ab⁴	ab⁵	b⁸	ab⁵	b⁹	ab⁶	b⁷	ab⁴	b⁸	ab⁵	ab⁶	ab⁴
b⁵	b⁵	ab⁴	b⁶	ab⁵	ab⁶	b⁷	b⁷	ab⁶	ab⁵	b⁸	b⁹	ab⁴	b⁹	ab⁵	b⁷	ab⁶	b⁸	ab⁴	b⁹	b⁷	b⁸
ab⁵	ab⁵	b⁷	ab⁶	b⁸	b⁹	ab⁴	ab⁴	b⁹	b⁸	ab⁵	ab⁶	b⁷	ab⁶	b⁸	ab⁴	b⁹	ab⁵	b⁷	ab⁶	ab⁴	ab⁵
b⁶	b⁶	ab⁶	b⁷	ab⁴	ab⁵	b⁸	b⁸	ab⁵	ab⁴	b⁹	b⁷	ab⁶	b⁷	ab⁴	b⁸	ab⁵	b⁹	ab⁶	b⁷	b⁸	b⁹
ab⁶	ab⁶	b⁹	ab⁴	b⁷	b⁸	ab⁵	ab⁵	b⁸	b⁷	ab⁶	ab⁴	b⁹	ab⁴	b⁷	ab⁵	b⁸	ab⁶	b⁹	ab⁴	ab⁵	ab⁶
b⁷	b⁷	ab⁵	b⁸	ab⁶	ab⁴	b⁹	b⁹	ab⁴	ab⁶	b⁷	b⁸	ab⁵	b⁸	ab⁶	b⁹	ab⁴	b⁷	ab⁵	b⁸	b⁹	b⁷
b⁸	b⁸	ab⁴	b⁹	ab⁵	ab⁶	b⁷	b⁷	ab⁶	ab⁵	b⁸	b⁹	ab⁴	b⁹	ab⁵	b⁷	ab⁶	b⁸	ab⁴	b⁹	b⁷	b⁸
b⁹	b⁹	ab⁶	b⁷	ab⁴	ab⁵	b⁸	b⁸	ab⁵	ab⁴	b⁹	b⁷	ab⁶	b⁷	ab⁴	b⁸	ab⁵	b⁹	ab⁶	b⁷	b⁸	b⁹

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.	b¹⁰ ⇒ b⁷	[7]
2.	ab⁷ ⇒ ab⁴	[6]
3.	bab ⇒ ab³	[4]
4.	b³a ⇒ ab³	[5]
5.	a² ⇒ b³	[2]

# ab:bab=aaa,bbb=aa b/a
bbbbbbbbbb=bbbbbbb
abbbbbbb=abbbb
bab=abbb
bbba=abbb
aa=bbb

Same cardinality

8 unique, 64 total

Σ	#	Presentation	Description	Related
8	402	⟨a, b \| aaa=ab, bbb=1⟩	Finite non-commutative monoid with 21 elements	12 iso, 27 anti-iso
9	1599	⟨a, b \| aaa=ab, bbb=b⟩	Finite non-commutative monoid with 21 elements	1 anti-iso
10	5309	⟨a, b \| aaa=aa, bbb=ab⟩	Finite non-commutative monoid with 21 elements
10	6265	⟨a, b \| aaa=a, abbbb=b⟩	Finite non-commutative monoid with 21 elements	2 iso
11	11127	⟨a, b \| aaaaa=b, abbbb=1⟩	Isomorphic to ℤ₂₁	13 iso
11	13091	⟨a, b \| bab=aaa, abba=b⟩	Finite non-commutative monoid with 21 elements
11	15158	⟨a, b \| aab=ba, ababab=1⟩	Finite non-Abelian group with 21 elements	1 iso
11	19212	⟨a, b \| aba=b, baaaab=a⟩	Finite non-commutative monoid with 21 elements

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