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〈a, b | aaa=b, bbb=a〉

Monoid presentation of length 8

Properties

Completion parameters

Complete rewriting system

  1. a9 ⇒ a
  2. b ⇒ a3

Idempotents

2 elements

Cayley table

Idempotents are shown in bold.

1aa2a3a4a5a6a7a8
11aa2a3a4a5a6a7a8
aaa2a3a4a5a6a7a8a
a2a2a3a4a5a6a7a8aa2
a3a3a4a5a6a7a8aa2a3
a4a4a5a6a7a8aa2a3a4
a5a5a6a7a8aa2a3a4a5
a6a6a7a8aa2a3a4a5a6
a7a7a8aa2a3a4a5a6a7
a8a8aa2a3a4a5a6a7a8

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

38 unique, 880 total

Length:Presentation:Description:Related:
7 〈a, b | aaa=b, bbb=1〉 Isomorphic to ℤ9 562 isomorphic
8 〈a, b | aaa=a, abb=b〉 Finite non-commutative monoid with 9 elements 2 isomorphic
8 〈a, b | aaa=a, bbb=a〉 Finite commutative monoid with 9 elements 48 isomorphic
8 〈a, b | aba=b, bab=a〉 Finite non-commutative monoid with 9 elements 3 isomorphic
9 〈a, b | aaa=aa, bbb=a〉 Finite commutative monoid with 9 elements 16 isomorphic
9 〈a, b | aaa=ab, bab=b〉 Finite non-commutative monoid with 9 elements 4 isomorphic
9 〈a, b | aaa=ab, bbb=a〉 Finite commutative monoid with 9 elements 40 isomorphic
9 〈a, b | aaa=bb, bbb=a〉 Finite commutative monoid with 9 elements 70 isomorphic
9 〈a, b | aaa=a, abab=b〉 Finite non-commutative monoid with 9 elements 2 isomorphic
9 〈a, b | aaa=a, abba=b〉 Finite non-commutative monoid with 9 elements 2 isomorphic
9 〈a, b | aab=b, abba=a〉 Finite non-commutative monoid with 9 elements 13 isomorphic
9 〈a, b | ab=aa, bba=bb〉 Finite non-commutative monoid with 9 elements
9 〈a, b | ab=a, baaaa=b〉 Finite non-commutative monoid with 9 elements 20 isomorphic
10 〈a, b | aab=aaa, baa=b〉 Finite non-commutative monoid with 9 elements 1 isomorphic
10 〈a, b | aab=aaa, bbb=a〉 Finite commutative monoid with 9 elements 1 isomorphic
10 〈a, b | abb=aaa, bbb=a〉 Finite commutative monoid with 9 elements 12 isomorphic
10 〈a, b | aaa=aa, abbb=b〉 Finite non-commutative monoid with 9 elements 3 isomorphic
10 〈a, b | aaa=bb, aaba=b〉 Finite commutative monoid with 9 elements
10 〈a, b | aab=aa, bbbb=a〉 Finite commutative monoid with 9 elements 5 isomorphic
10 〈a, b | aaa=bb, aab=ab〉 Finite non-commutative monoid with 9 elements 1 isomorphic
10 〈a, b | aaa=a, ababa=b〉 Finite non-commutative monoid with 9 elements
10 〈a, b | aab=a, baaaa=b〉 Finite non-commutative monoid with 9 elements
10 〈a, b | aba=a, aaaab=b〉 Finite non-commutative monoid with 9 elements
10 〈a, b | aab=b, bbb=aaa〉 Finite commutative monoid with 9 elements 1 isomorphic
10 〈a, b | bb=aa, aaaa=ab〉 Finite non-commutative monoid with 9 elements 2 isomorphic
10 〈a, b | bb=aa, aabb=ba〉 Finite non-commutative monoid with 9 elements
10 〈a, b | ab=a, baaa=bbb〉 Finite non-commutative monoid with 9 elements 3 isomorphic
10 〈a, b | ab=a, bbaa=bbb〉 Finite non-commutative monoid with 9 elements 2 isomorphic
11 〈a, b | aba=aaa, aaab=b〉 Finite non-commutative monoid with 9 elements 3 isomorphic
11 〈a, b | abb=aaa, aaab=b〉 Finite commutative monoid with 9 elements 7 isomorphic
11 〈a, b | aaab=a, baaaa=b〉 Finite non-commutative monoid with 9 elements 1 isomorphic
11 〈a, b | aab=b, aaaaa=ba〉 Finite non-commutative monoid with 9 elements
11 〈a, b | aab=b, aaaaa=bb〉 Finite commutative monoid with 9 elements
11 〈a, b | aab=b, abaa=aaa〉 Finite non-commutative monoid with 9 elements
11 〈a, b | aab=b, babb=aaa〉 Finite commutative monoid with 9 elements 3 isomorphic
11 〈a, b | aba=a, abab=bbb〉 Finite non-commutative monoid with 9 elements
11 〈a, b | ab=aa, aaaaa=bb〉 Finite non-commutative monoid with 9 elements 15 isomorphic
11 〈a, b | ab=a, bbbbb=baa〉 Finite non-commutative monoid with 9 elements

Other isomorphic instances

62 total

Length:Presentation:
9〈a, b | aab=b, bbbb=a〉
9〈a, b | aba=b, bbbb=a〉
9〈a, b | aa=b, abbbb=a〉
9〈a, b | aa=b, babbb=a〉
9〈a, b | aa=b, bbabb=a〉
10〈a, b | aaa=b, aaabb=a〉
10〈a, b | aaa=b, aabab=a〉
10〈a, b | aaa=b, aabba=a〉
10〈a, b | aaa=b, abaab=a〉
10〈a, b | aaa=b, ababa=a〉
10〈a, b | aaa=b, baaab=a〉
10〈a, b | aa=b, aaabbb=a〉
10〈a, b | aa=b, aababb=a〉
10〈a, b | aa=b, aabbab=a〉
10〈a, b | aa=b, aabbba=a〉
10〈a, b | aa=b, abaabb=a〉
10〈a, b | aa=b, ababab=a〉
10〈a, b | aa=b, ababba=a〉
10〈a, b | aa=b, abbaab=a〉
10〈a, b | aa=b, baaabb=a〉
10〈a, b | aa=b, baabab=a〉
11〈a, b | aabb=a, aaaab=b〉
11〈a, b | aabb=a, aaaba=b〉
11〈a, b | aabb=a, aabaa=b〉
11〈a, b | aabb=a, abaaa=b〉
11〈a, b | abab=a, aaaab=b〉
11〈a, b | abab=a, aaaba=b〉
11〈a, b | abab=a, aabaa=b〉
11〈a, b | abab=a, abaaa=b〉
11〈a, b | abba=a, aaaba=b〉
11〈a, b | abba=a, aabaa=b〉
11〈a, b | abba=b, abbbb=a〉
11〈a, b | abba=b, babbb=a〉
11〈a, b | aab=b, aabbbb=a〉
11〈a, b | aab=b, ababbb=a〉
11〈a, b | aab=b, abbabb=a〉
11〈a, b | aab=b, abbbab=a〉
11〈a, b | aab=b, baabbb=a〉
11〈a, b | aab=b, bababb=a〉
11〈a, b | aab=b, babbab=a〉
11〈a, b | aab=b, bbaabb=a〉
11〈a, b | aab=b, bbabab=a〉
11〈a, b | aab=b, bbbaab=a〉
11〈a, b | aba=b, aabbbb=a〉
11〈a, b | aba=b, ababbb=a〉
11〈a, b | aba=b, abbabb=a〉
11〈a, b | aba=b, abbbab=a〉
11〈a, b | aba=b, abbbba=a〉
11〈a, b | aba=b, bababb=a〉
11〈a, b | aba=b, bbaabb=a〉
11〈a, b | aa=b, aaaaabb=a〉
11〈a, b | aa=b, aaaabab=a〉
11〈a, b | aa=b, aaaabba=a〉
11〈a, b | aa=b, aaabaab=a〉
11〈a, b | aa=b, aaababa=a〉
11〈a, b | aa=b, aaabbaa=a〉
11〈a, b | aa=b, aabaaab=a〉
11〈a, b | aa=b, aabaaba=a〉
11〈a, b | aa=b, aababaa=a〉
11〈a, b | aa=b, abaaaab=a〉
11〈a, b | aa=b, abaaaba=a〉
11〈a, b | aa=b, baaaaab=a〉