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〈a, b | aaaa=b, bbbb=a〉

Monoid presentation of length 10

Properties

Finite commutative monoid with 16 elements
Isomorphic to commutative presentation: 〈a | 16a=a〉
Grothendieck group is ℤ₁₅

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a¹⁶ ⇒ a
b ⇒ a⁴

Idempotents

2 elements

1
a¹⁵

Cayley table

Idempotents are shown in bold.

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

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 170 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| bb=aa, abab=1〉	Finite non-Abelian group with 16 elements	4 isomorphic
8	〈a, b \| aa=1, ababbb=1〉	Finite non-Abelian group with 16 elements	40 isomorphic
9	〈a, b \| aaaa=b, bbbb=1〉	Isomorphic to ℤ₁₆	67 isomorphic
9	〈a, b \| aab=a, bbbb=b〉	Finite non-commutative monoid with 16 elements
10	〈a, b \| aaab=ba, abab=1〉	Finite non-Abelian group with 16 elements	7 isomorphic
10	〈a, b \| bab=aaa, aabb=1〉	Finite non-Abelian group with 16 elements	6 isomorphic
10	〈a, b \| aaaa=a, aabb=b〉	Finite non-commutative monoid with 16 elements	1 isomorphic
10	〈a, b \| aaaa=a, bbbb=a〉	Finite commutative monoid with 16 elements	1 isomorphic
10	〈a, b \| aba=b, aaaabb=1〉	Finite non-Abelian group with 16 elements	3 isomorphic
10	〈a, b \| aba=b, aabbbb=1〉	Finite non-Abelian group with 16 elements	5 isomorphic
11	〈a, b \| aaaa=aa, bbbb=a〉	Finite commutative monoid with 16 elements
11	〈a, b \| aaaa=ab, bbbb=a〉	Finite commutative monoid with 16 elements	2 isomorphic
11	〈a, b \| aaaa=bb, bbbb=a〉	Finite commutative monoid with 16 elements
11	〈a, b \| aaab=bb, abba=a〉	Finite non-commutative monoid with 16 elements	2 isomorphic
11	〈a, b \| bab=aab, bbb=aa〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| bab=aba, bbb=aa〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aaaa=a, aabab=b〉	Finite non-commutative monoid with 16 elements	1 isomorphic
11	〈a, b \| aab=bb, abbba=a〉	Finite non-commutative monoid with 16 elements	3 isomorphic
11	〈a, b \| aaa=ab, babb=bb〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aaa=ab, bbaa=bb〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aaa=bb, abab=aa〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aba=aa, bbbb=ab〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aaa=b, abbbbb=b〉	Finite commutative monoid with 16 elements	2 isomorphic
11	〈a, b \| aba=b, aaaa=abb〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| ab=aa, bbba=bbb〉	Finite non-commutative monoid with 16 elements
11	〈a, b \| aa=a, ababab=bb〉	Finite non-commutative monoid with 16 elements

Other isomorphic instances

5 total

Length:	Presentation:
11	〈a, b \| aaab=b, bbbbb=a〉
11	〈a, b \| aaba=b, bbbbb=a〉
11	〈a, b \| aaa=b, abbbbb=a〉
11	〈a, b \| aaa=b, babbbb=a〉
11	〈a, b \| aaa=b, bbabbb=a〉