〈a, b | aaaa=b, abbbb=a〉

Monoid presentation of length 11

Properties

Finite commutative monoid with 17 elements
Isomorphic to commutative presentation: 〈a | 17a=a〉
Grothendieck group is ℤ₁₆

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a¹⁷ ⇒ a
b ⇒ a⁴

Idempotents

2 elements

1
a¹⁶

Cayley table

Idempotents are shown in bold.

	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴
a⁵	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵
a⁶	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶
a⁷	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷
a⁸	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸
a⁹	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹
a¹⁰	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
a¹¹	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹
a¹²	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²
a¹³	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³
a¹⁴	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴
a¹⁵	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵
a¹⁶	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

19 unique, 69 total

Length:	Presentation:	Description:	Related:
10	〈a, b \| aaaa=b, abbbb=1〉	Isomorphic to ℤ₁₇	39 isomorphic
10	〈a, b \| aab=b, babbb=a〉	Finite non-commutative monoid with 17 elements	1 isomorphic
10	〈a, b \| aba=b, baaab=a〉	Finite non-commutative monoid with 17 elements
10	〈a, b \| aba=b, babbb=a〉	Finite non-commutative monoid with 17 elements
10	〈a, b \| bb=aa, abab=aa〉	Finite non-commutative monoid with 17 elements
10	〈a, b \| bb=aa, abab=ab〉	Finite non-commutative monoid with 17 elements
10	〈a, b \| bb=aa, abab=ba〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aab=aaa, bbb=ab〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aab=aaa, bbb=ba〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aba=aaa, baa=bb〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aba=aaa, bbb=ab〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| abb=aba, bbb=aa〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aaaa=b, abbbb=b〉	Finite commutative monoid with 17 elements	2 isomorphic
11	〈a, b \| aabb=a, baaaa=b〉	Finite non-commutative monoid with 17 elements	1 isomorphic
11	〈a, b \| abba=a, aaaab=b〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aaa=ab, babbb=b〉	Finite non-commutative monoid with 17 elements	3 isomorphic
11	〈a, b \| aab=aa, bbbb=ba〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aab=bb, bab=aaa〉	Finite non-commutative monoid with 17 elements
11	〈a, b \| aab=b, abbbba=a〉	Finite non-commutative monoid with 17 elements	4 isomorphic

Other isomorphic instances

2 total

Length:	Presentation:
11	〈a, b \| aaaa=b, babbb=a〉
11	〈a, b \| aaaa=b, bbabb=a〉