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〈a, b | aaaa=a, bbbb=a〉

Monoid presentation of length 10

Properties

Finite commutative monoid with 16 elements
Isomorphic to commutative presentation: 〈a | 16a=4a〉
Grothendieck group is ℤ₁₂

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b¹⁶ ⇒ b⁴
a ⇒ b⁴

Idempotents

2 elements

1
b¹²

Cayley table

Idempotents are shown in bold.

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

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 174 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=b, bbbb=a〉	Finite commutative monoid with 16 elements	5 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

1 total

Length:	Presentation:
11	〈a, b \| aaaa=bb, bbbb=b〉