#19374 ⟨a, b | aaa=b, bbbbb=ab⟩

Properties

Presentation has sum-of-sides 11
Isomorphic to ℕ_{(15 = 4)}
Finite commutative monoid with 15 elements
Commutative Gröbner basis: ⟨a, b | a¹⁵=a⁴, b=a³⟩
Not cancellative, because multiplication by a⁴ is not injective:
- a¹² ≠ a
- a⁴ ⋅ a¹² = a⁵
- a⁴ ⋅ a = a⁵
Cancellative quotient is isomorphic to ℤ₁₁
Grothendieck group is isomorphic to ℤ₁₁
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

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 non-trivial idempotent:
- a¹¹
Order of generators:
- a: index 4, period 11
- b: index 2, period 11
Histogram:

index 1, period 1 1 element a¹¹

index 1, period 11 10 elements a⁴, a⁵, a⁶, a⁷, a⁸, ...

index 2, period 11 2 elements a², a³

index 4, period 11 1 element a

index 1, period 1	1 element	a¹¹
index 1, period 11	10 elements	a⁴, a⁵, a⁶, a⁷, a⁸, ...
index 2, period 11	2 elements	a², a³
index 4, period 11	1 element	a

Complete 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

a¹⁵ ⇒ a⁴
b ⇒ a³

# ab:aaa=b,bbbbb=ab a/b
aaaaaaaaaaaaaaa=aaaa
b=aaa

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴	a⁵
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴	a⁵	a⁶
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴	a⁵	a⁶	a⁷
a⁵	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴	a⁵	a⁶	a⁷	a⁸
a⁶	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹
a⁷	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
a⁸	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹
a⁹	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²
a¹⁰	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³
a¹¹	a¹¹	a¹²	a¹³	a¹⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴
a¹²	a¹²	a¹³	a¹⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴
a¹³	a¹³	a¹⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴	a⁵
a¹⁴	a¹⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a⁴	a⁵	a⁶

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

22 unique, 190 total

Σ	#	Presentation	Description	Related
9	1602	⟨a, b \| aaa=bb, aba=a⟩	Finite non-commutative monoid with 15 elements	2 iso
9	1603	⟨a, b \| aaa=bb, aba=b⟩	Finite non-commutative monoid with 15 elements	1 iso
9	1751	⟨a, b \| aaab=1, bbbbb=1⟩	Isomorphic to ℤ₁₅	138 iso
9	1993	⟨a, b \| aaa=a, abbb=b⟩	Finite non-commutative monoid with 15 elements	4 iso
10	5342	⟨a, b \| aaa=bb, aba=aa⟩	Finite non-commutative monoid with 15 elements	1 iso
10	5344	⟨a, b \| aaa=bb, aba=bb⟩	Finite non-commutative monoid with 15 elements	2 iso
10	6276	⟨a, b \| aaa=a, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 5)}
10	6316	⟨a, b \| aaa=b, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 1)}
10	6317	⟨a, b \| aaa=b, bbbbb=b⟩	Isomorphic to ℕ_{(15 = 3)}
11	12981	⟨a, b \| abb=aaa, baaa=b⟩	Finite non-commutative monoid with 15 elements	3 iso
11	12985	⟨a, b \| abb=aaa, baba=b⟩	Finite non-commutative monoid with 15 elements	7 iso
11	13123	⟨a, b \| aab=aaa, aba=bb⟩	Finite non-commutative monoid with 15 elements
11	13191	⟨a, b \| abb=aaa, bbb=ab⟩	Finite non-commutative monoid with 15 elements	1 iso
11	13192	⟨a, b \| abb=aaa, bbb=ba⟩	Finite non-commutative monoid with 15 elements
11	15439	⟨a, b \| aaa=aa, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 10)}
11	15503	⟨a, b \| aaa=ab, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 6)}	1 iso
11	15543	⟨a, b \| aaa=bb, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 2)}	1 iso
11	16041	⟨a, b \| aaa=ab, bbbb=aa⟩	Finite non-commutative monoid with 15 elements
11	16142	⟨a, b \| aab=aa, bbbb=ab⟩	Finite non-commutative monoid with 15 elements
11	19502	⟨a, b \| aab=a, bbbbb=ba⟩	Finite non-commutative monoid with 15 elements	1 anti-iso
11	20144	⟨a, b \| aab=b, bbaa=aaa⟩	Finite non-commutative monoid with 15 elements
11	20819	⟨a, b \| ab=aa, bbaaa=bb⟩	Finite non-commutative monoid with 15 elements	6 iso

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