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〈a, b | aa=b, bb=a〉

Monoid presentation of length 6

Properties

Completion parameters

Complete rewriting system

  1. a4 ⇒ a
  2. b ⇒ a2

Idempotents

2 elements

Cayley table

Idempotents are shown in bold.

1aa2a3
11aa2a3
aaa2a3a
a2a2a3aa2
a3a3aa2a3

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

8 unique, 1535 total

Length:Presentation:Description:Related:
5 〈a, b | aa=b, bb=1〉 Isomorphic to ℤ4 1419 isomorphic
6 〈a, b | aa=a, bb=a〉 Finite commutative monoid with 4 elements 37 isomorphic
7 〈a, b | ab=aa, ba=b〉 Finite non-commutative monoid with 4 elements 8 isomorphic, 6 anti-isomorphic
7 〈a, b | ab=aa, bb=a〉 Finite commutative monoid with 4 elements 16 isomorphic
7 〈a, b | aa=a, abb=b〉 Finite non-commutative monoid with 4 elements 14 isomorphic
7 〈a, b | ab=a, bb=aa〉 Finite commutative monoid with 4 elements 17 isomorphic
9 〈a, b | aa=a, abbba=b〉 Finite commutative monoid with 4 elements 8 isomorphic
10 〈a, b | aaa=aa, abba=b〉 Finite commutative monoid with 4 elements 2 isomorphic

Other isomorphic instances

72 total

Length:Presentation:
7〈a, b | aa=b, aab=a〉
7〈a, b | aa=b, aba=a〉
7〈a, b | ab=a, aaa=b〉
8〈a, b | aa=b, aaaa=a〉
8〈a, b | ab=a, aaab=b〉
8〈a, b | ab=a, aaba=b〉
8〈a, b | ab=a, abaa=b〉
9〈a, b | aaa=b, aaaa=a〉
9〈a, b | aab=a, aaab=b〉
9〈a, b | aab=a, aaba=b〉
9〈a, b | aab=a, abaa=b〉
9〈a, b | aba=a, aaba=b〉
9〈a, b | ab=a, aaabb=b〉
9〈a, b | ab=a, aabab=b〉
9〈a, b | ab=a, aabba=b〉
9〈a, b | ab=a, abaab=b〉
9〈a, b | ab=a, ababa=b〉
9〈a, b | ab=a, abbaa=b〉
10〈a, b | aaab=a, aaba=b〉
10〈a, b | aaab=a, abaa=b〉
10〈a, b | aaab=a, abbb=b〉
10〈a, b | aaba=a, abaa=b〉
10〈a, b | aabb=a, abab=b〉
10〈a, b | aabb=a, abba=b〉
10〈a, b | abab=a, abba=b〉
10〈a, b | ab=a, aaabbb=b〉
10〈a, b | ab=a, aababb=b〉
10〈a, b | ab=a, aabbab=b〉
10〈a, b | ab=a, aabbba=b〉
10〈a, b | ab=a, abaabb=b〉
10〈a, b | ab=a, ababab=b〉
10〈a, b | ab=a, ababba=b〉
10〈a, b | ab=a, abbaab=b〉
10〈a, b | ab=a, abbaba=b〉
10〈a, b | ab=a, abbbaa=b〉
11〈a, b | aaaa=a, aaaaa=b〉
11〈a, b | aaab=b, aaaab=a〉
11〈a, b | aaba=b, aaaab=a〉
11〈a, b | aaba=b, aaaba=a〉
11〈a, b | aaba=b, aabaa=a〉
11〈a, b | aaba=b, abaaa=a〉
11〈a, b | aaba=b, baaaa=a〉
11〈a, b | aab=a, aaaabb=b〉
11〈a, b | aab=a, aaabab=b〉
11〈a, b | aab=a, aaabba=b〉
11〈a, b | aab=a, aabaab=b〉
11〈a, b | aab=a, aababa=b〉
11〈a, b | aab=a, aabbaa=b〉
11〈a, b | aab=a, abaaab=b〉
11〈a, b | aab=a, abaaba=b〉
11〈a, b | aab=a, ababaa=b〉
11〈a, b | aab=a, abbaaa=b〉
11〈a, b | aab=a, abbbbb=b〉
11〈a, b | aba=a, aaabba=b〉
11〈a, b | aba=a, aababa=b〉
11〈a, b | aba=a, aabbaa=b〉
11〈a, b | aba=a, abaaba=b〉
11〈a, b | ab=a, aaabbbb=b〉
11〈a, b | ab=a, aababbb=b〉
11〈a, b | ab=a, aabbabb=b〉
11〈a, b | ab=a, aabbbab=b〉
11〈a, b | ab=a, aabbbba=b〉
11〈a, b | ab=a, abaabbb=b〉
11〈a, b | ab=a, abababb=b〉
11〈a, b | ab=a, ababbab=b〉
11〈a, b | ab=a, ababbba=b〉
11〈a, b | ab=a, abbaabb=b〉
11〈a, b | ab=a, abbabab=b〉
11〈a, b | ab=a, abbabba=b〉
11〈a, b | ab=a, abbbaab=b〉
11〈a, b | ab=a, abbbaba=b〉
11〈a, b | ab=a, abbbbaa=b〉