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〈a, b | aaaa=a, aabb=b〉

Monoid presentation of length 10

Properties

Completion parameters

Complete rewriting system

  1. ab ⇒ b2
  2. b4 ⇒ b
  3. a4 ⇒ a

Idempotents

6 elements

Cayley table

Idempotents are shown in bold.

1aba2bab2a3ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
11aba2bab2a3ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
aaa2b2a3b2ab3ab2a2b3abb2a3b3a2bab3a3ba2ba3
bbbab2ba2b2ab3ba3b2a2b3abb2a3b3a2bab3a3ba2ba3
a2a2a3b3ab3aba2b3a2bab2b3a3ba2b2aba3b2a2b2a3
bababa2b3ba3b3abbab3a2bab2b3a3ba2b2aba3b2a2b2a3
b2b2b2ab3b2a2b3abb2a3b3a2bab2b3a3ba2b2aba3b2a2b2a3
a3a3aba2bab2a3ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
ba2ba2ba3bbabab2ba2ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
b2ab2ab2a2bb2a3bab2b2aba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
b3b3b3abb3a2bab2b3a3ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
ba3ba3bab2ba2b2ab3ba3b2a2b3abb2a3b3a2bab3a3ba2ba3
b2a2b2a2b2a3b2b2ab2ab3b2a2b2a2b3abb2a3b3a2bab3a3ba2ba3
b3ab3ab3a2b2b3a3b2ab3b3ab2a2b3abb2a3b3a2bab3a3ba2ba3
b2a3b2a3b2ab3b2a2b3abb2a3b3a2bab2b3a3ba2b2aba3b2a2b2a3
b3a2b3a2b3a3b3b3ab3abb3a2b3a2bab2b3a3ba2b2aba3b2a2b2a3
b3a3b3a3b3abb3a2bab2b3a3ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3

Right Cayley graph

Idempotents are shown in bold.

Left 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, bbbb=a〉 Finite 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:
10〈a, b | aaaa=a, abab=b〉