#18758 ⟨a, b | aaa=a, bbbbbb=a⟩

Properties

Presentation has sum-of-sides 11
Isomorphic to ℕ_{(18 = 6)}
Finite commutative monoid with 18 elements
Commutative Gröbner basis: ⟨a, b | b¹⁸=b⁶, a=b⁶⟩
Not cancellative, because multiplication by b⁷ is not injective:
- b¹² ≠ 1
- b⁷ ⋅ b¹² = b⁷
- b⁷ ⋅ 1 = b⁷
- Cancellative quotient is isomorphic to ℤ₁₂
Grothendieck group is isomorphic to ℤ₁₂
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, b¹²

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(b) = 0; deg(a) = 1

b¹⁸ ⇒ b⁶
a ⇒ b⁶

# ab:aaa=a,bbbbbb=a b/a
bbbbbbbbbbbbbbbbbb=bbbbbb
a=bbbbbb

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷
1	1	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷
b	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶
b²	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷
b³	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸
b⁴	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹
b⁵	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰
b⁶	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹
b⁷	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²
b⁸	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³
b⁹	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴
b¹⁰	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵
b¹¹	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶
b¹²	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷
b¹³	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶
b¹⁴	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷
b¹⁵	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸
b¹⁶	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹
b¹⁷	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰

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	18830	⟨a, b \| aaa=b, bbbbbb=a⟩	Isomorphic to ℕ_{(18 = 1)}
11	18831	⟨a, b \| aaa=b, bbbbbb=b⟩	Isomorphic to ℕ_{(18 = 3)}
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

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