#19212 ⟨a, b | aba=b, baaaab=a⟩

Quick links

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 a is not injective:
- a ⋅ a¹⁰ = a and a ⋅ 1 = a, 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 1, 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	a, a³, a⁷, a⁹

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
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹
a	a	a²	ba⁹	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a	ba⁸
b	b	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a	ba⁸	a²	ba⁹	a³	b	a⁴
a²	a²	a³	ba⁸	a⁴	ba⁹	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a	ba⁶	a²	ba⁷
ba	ba	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a	b	a²	ba	a³
a³	a³	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba⁹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a	ba⁴	a²	ba⁵	a³	ba⁶
ba²	ba²	ba³	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba⁹	a⁹	b	a¹⁰	ba	a	ba²	a²
a⁴	a⁴	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba⁹	a⁹	b	a¹⁰	ba	a	ba²	a²	ba³	a³	ba⁴	a⁴	ba⁵
ba³	ba³	ba⁴	a²	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba⁹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a
a⁵	a⁵	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴
ba⁴	ba⁴	ba⁵	a	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰
a⁶	a⁶	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a	ba⁸	a²	ba⁹	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³
ba⁵	ba⁵	ba⁶	a¹⁰	ba⁷	a	ba⁸	a²	ba⁹	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹
a⁷	a⁷	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	b	a⁶	ba	a⁷	ba²
ba⁶	ba⁶	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸
a⁸	a⁸	a⁹	ba²	a¹⁰	ba³	a	ba⁴	a²	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba⁹	a⁷	b	a⁸	ba
ba⁷	ba⁷	ba⁸	a⁸	ba⁹	a⁹	b	a¹⁰	ba	a	ba²	a²	ba³	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷
a⁹	a⁹	a¹⁰	ba	a	ba²	a²	ba³	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba⁹	a⁹	b
ba⁸	ba⁸	ba⁹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a	ba⁴	a²	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶
a¹⁰	a¹⁰	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹
ba⁹	ba⁹	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵

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.	a¹¹ ⇒ a	[13]
2.	ba¹⁰ ⇒ b	[12]
3.	ab ⇒ ba⁹	[9]
4.	b² ⇒ a⁵	[8]

# ab:aba=b,baaaab=a a/b
aaaaaaaaaaa=a
baaaaaaaaaa=b
ab=baaaaaaaaa
bb=aaaaa

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	13248	⟨a, b \| bab=aaa, bbb=aa⟩	Finite non-commutative monoid with 21 elements
11	15158	⟨a, b \| aab=ba, ababab=1⟩	Finite non-Abelian group with 21 elements	1 iso

Up:	Monoids with two generators and two relations
Prev:	#19211 ⟨a, b \| aba=b, abbbbb=b⟩
Next:	#19231 ⟨a, b \| aba=b, bbbbbb=b⟩