#15439 ⟨a, b | aaa=aa, bbbbb=a⟩

Properties

Presentation has sum-of-sides 11
Isomorphic to ℕ_{(15 = 10)}
Finite commutative monoid with 15 elements
Commutative Gröbner basis: ⟨a, b | b¹⁵=b¹⁰, a=b⁵⟩
Not cancellative, because multiplication by b¹² is not injective:
- b⁵ ≠ 1
- b¹² ⋅ b⁵ = b¹²
- b¹² ⋅ 1 = b¹²
- Cancellative quotient is isomorphic to ℤ₅
Grothendieck group is isomorphic to ℤ₅
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, b¹⁰

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

b¹⁵ ⇒ b¹⁰
a ⇒ b⁵

# ab:aaa=aa,bbbbb=a b/a
bbbbbbbbbbbbbbb=bbbbbbbbbb
a=bbbbb

Staircase diagram

Cayley table

Idempotents are shown in bold.

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

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	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	19374	⟨a, b \| aaa=b, bbbbb=ab⟩	Isomorphic to ℕ_{(15 = 4)}
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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