#18831 ⟨a, b | aaa=b, bbbbbb=b⟩

Properties

Presentation has sum-of-sides 11
Isomorphic to ℕ_{(18 = 3)}
Finite commutative monoid with 18 elements
Commutative Gröbner basis: ⟨a, b | a¹⁸=a³, b=a³⟩
Not cancellative, because multiplication by a⁴ is not injective:
- a¹⁵ ≠ 1
- a⁴ ⋅ a¹⁵ = a⁴
- a⁴ ⋅ 1 = a⁴
- Cancellative quotient is isomorphic to ℤ₁₅
Grothendieck group is isomorphic to ℤ₁₅
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, a¹⁵

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

a¹⁸ ⇒ a³
b ⇒ a³

# ab:aaa=b,bbbbbb=b a/b
aaaaaaaaaaaaaaaaaa=aaa
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	a¹⁵	a¹⁶	a¹⁷
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶
a⁵	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷
a⁶	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷	a⁸
a⁷	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹
a⁸	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
a⁹	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹
a¹⁰	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²
a¹¹	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³
a¹²	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴
a¹³	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵
a¹⁴	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶
a¹⁵	a¹⁵	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷
a¹⁶	a¹⁶	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³
a¹⁷	a¹⁷	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a³	a⁴

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

17 unique, 104 total

Σ	#	Presentation	Description	Related
8	1121	⟨a, b \| aa=1, abbba=b⟩	Finite non-commutative monoid with 18 elements	17 iso
9	3387	⟨a, b \| aa=1, abbbbb=b⟩	Finite non-commutative monoid with 18 elements	22 iso, 9 anti-iso
10	5521	⟨a, b \| aaab=1, bbbbbb=1⟩	Isomorphic to ℤ₁₈	33 iso
10	6732	⟨a, b \| aba=a, aaab=bb⟩	Finite non-commutative monoid with 18 elements
10	6788	⟨a, b \| aba=b, baab=aa⟩	Finite non-commutative monoid with 18 elements
10	6795	⟨a, b \| aba=b, bbbb=aa⟩	Finite non-commutative monoid with 18 elements
10	7039	⟨a, b \| bb=aa, aaaba=b⟩	Finite non-commutative monoid with 18 elements	2 iso
10	8910	⟨a, b \| aa=a, bbbbb=ab⟩	Finite non-commutative monoid with 18 elements	1 iso
11	12187	⟨a, b \| aaaa=ab, babb=b⟩	Finite non-commutative monoid with 18 elements	2 iso
11	15797	⟨a, b \| aab=bb, bbbba=a⟩	Finite non-commutative monoid with 18 elements
11	16042	⟨a, b \| aaa=ab, bbbb=ab⟩	Finite non-commutative monoid with 18 elements
11	16043	⟨a, b \| aaa=ab, bbbb=ba⟩	Finite non-commutative monoid with 18 elements
11	16313	⟨a, b \| aab=bb, baba=aa⟩	Finite non-commutative monoid with 18 elements
11	18758	⟨a, b \| aaa=a, bbbbbb=a⟩	Isomorphic to ℕ_{(18 = 6)}
11	18830	⟨a, b \| aaa=b, bbbbbb=a⟩	Isomorphic to ℕ_{(18 = 1)}
11	19624	⟨a, b \| aab=b, bbbba=aa⟩	Finite non-commutative monoid with 18 elements
11	20914	⟨a, b \| bb=aa, ababa=aa⟩	Finite non-commutative monoid with 18 elements	1 iso

Up:	Monoids with two generators and two relations
Prev:	#18830 ⟨a, b \| aaa=b, bbbbbb=a⟩
Next:	#18832 ⟨a, b \| aab=a, aaaaaa=a⟩