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〈a, b | aa=1, abbbbb=b〉

Monoid presentation of length 9

Properties

Completion parameters

Complete rewriting system

  1. b9 ⇒ b
  2. ab ⇒ b5
  3. a2 ⇒ 1

Idempotents

3 elements

Cayley table

Idempotents are shown in bold.

1abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8a
11abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8a
aa1b5b5ab6b6ab7b7ab8b8abbab2b2ab3b3ab4b4a
bbbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8abba
bababb6b6ab7b7ab8b8abbab2b2ab3b3ab4b4ab5b5a
b2b2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8abbab2b2a
b2ab2ab2b7b7ab8b8abbab2b2ab3b3ab4b4ab5b5ab6b6a
b3b3b3ab4b4ab5b5ab6b6ab7b7ab8b8abbab2b2ab3b3a
b3ab3ab3b8b8abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7a
b4b4b4ab5b5ab6b6ab7b7ab8b8abbab2b2ab3b3ab4b4a
b4ab4ab4bbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8a
b5b5b5ab6b6ab7b7ab8b8abbab2b2ab3b3ab4b4ab5b5a
b5ab5ab5b2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8abba
b6b6b6ab7b7ab8b8abbab2b2ab3b3ab4b4ab5b5ab6b6a
b6ab6ab6b3b3ab4b4ab5b5ab6b6ab7b7ab8b8abbab2b2a
b7b7b7ab8b8abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7a
b7ab7ab7b4b4ab5b5ab6b6ab7b7ab8b8abbab2b2ab3b3a
b8b8b8abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8a
b8ab8ab8b5b5ab6b6ab7b7ab8b8abbab2b2ab3b3ab4b4a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

18 unique, 73 total

Length:Presentation:Description:Related:
8 〈a, b | aa=1, abbba=b〉 Finite non-commutative monoid with 18 elements 14 isomorphic
10 〈a, b | aaab=1, bbbbbb=1〉 Isomorphic to ℤ18 33 isomorphic
10 〈a, b | aba=a, aaab=bb〉 Finite non-commutative monoid with 18 elements
10 〈a, b | aba=b, baab=aa〉 Finite non-commutative monoid with 18 elements
10 〈a, b | aba=b, bbbb=aa〉 Finite non-commutative monoid with 18 elements
10 〈a, b | bb=aa, aaaba=b〉 Finite non-commutative monoid with 18 elements 2 isomorphic
10 〈a, b | aa=a, bbbbb=ab〉 Finite non-commutative monoid with 18 elements 1 isomorphic
10 〈a, b | aa=1, abababa=b〉 Finite non-commutative monoid with 18 elements 2 isomorphic
11 〈a, b | aaaa=ab, babb=b〉 Finite non-commutative monoid with 18 elements 2 isomorphic
11 〈a, b | aab=bb, bbbba=a〉 Finite non-commutative monoid with 18 elements
11 〈a, b | aaa=ab, bbbb=ab〉 Finite non-commutative monoid with 18 elements
11 〈a, b | aaa=ab, bbbb=ba〉 Finite non-commutative monoid with 18 elements
11 〈a, b | aab=bb, baba=aa〉 Finite non-commutative monoid with 18 elements
11 〈a, b | aaa=a, bbbbbb=a〉 Finite commutative monoid with 18 elements
11 〈a, b | aaa=b, bbbbbb=a〉 Finite commutative monoid with 18 elements
11 〈a, b | aaa=b, bbbbbb=b〉 Finite commutative monoid with 18 elements
11 〈a, b | aab=b, bbbba=aa〉 Finite non-commutative monoid with 18 elements
11 〈a, b | bb=aa, ababa=aa〉 Finite non-commutative monoid with 18 elements 1 isomorphic

Other isomorphic instances

22 total

Length:Presentation:
9〈a, b | aa=1, bbbbb=ab〉
11〈a, b | aa=1, aaabbbbb=b〉
11〈a, b | aa=1, abaabbbb=b〉
11〈a, b | aa=1, abababbb=b〉
11〈a, b | aa=1, ababbabb=b〉
11〈a, b | aa=1, ababbbab=b〉
11〈a, b | aa=1, abbaabbb=b〉
11〈a, b | aa=1, abbababb=b〉
11〈a, b | aa=1, abbabbab=b〉
11〈a, b | aa=1, abbbaabb=b〉
11〈a, b | aa=1, abbbabab=b〉
11〈a, b | aa=1, abbbbaab=b〉
11〈a, b | aa=1, aabbbbb=ab〉
11〈a, b | aa=1, baabbbb=ab〉
11〈a, b | aa=1, bababbb=ab〉
11〈a, b | aa=1, babbabb=ab〉
11〈a, b | aa=1, babbbab=ab〉
11〈a, b | aa=1, bbaabbb=ab〉
11〈a, b | aa=1, bbababb=ab〉
11〈a, b | aa=1, abbbbb=aab〉
11〈a, b | aa=1, abbbbb=baa〉
11〈a, b | aa=1, bbbbb=aaab〉

Other anti-isomorphic instances

9 total

Length:Presentation:
11〈a, b | aa=1, aabbbbba=b〉
11〈a, b | aa=1, aabbbbb=ba〉
11〈a, b | aa=1, abbbbba=ab〉
11〈a, b | aa=1, baabbbb=ba〉
11〈a, b | aa=1, bababbb=ba〉
11〈a, b | aa=1, babbabb=ba〉
11〈a, b | aa=1, bbaabbb=ba〉
11〈a, b | aa=1, abbbbb=aba〉
11〈a, b | aa=1, bbbbb=aaba〉