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〈a, b | aa=1, abbba=b〉

Monoid presentation of length 8

Properties

Completion parameters

Complete rewriting system

  1. b9 ⇒ b
  2. ba ⇒ ab3
  3. a2 ⇒ 1

Idempotents

2 elements

Cayley table

Idempotents are shown in bold.

1ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
11ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
aa1abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8
bbab3b2ab4b3ab5b4ab6b5ab7b6ab8b7abb8ab2bab3
ababb3ab2b4ab3b5ab4b6ab5b7ab6b8ab7bab8b2abb3
b2b2ab6b3ab7b4ab8b5abb6ab2b7ab3b8ab4bab5b2ab6
ab2ab2b6ab3b7ab4b8ab5bab6b2ab7b3ab8b4abb5ab2b6
b3b3abb4ab2b5ab3b6ab4b7ab5b8ab6bab7b2ab8b3ab
ab3ab3bab4b2ab5b3ab6b4ab7b5ab8b6abb7ab2b8ab3b
b4b4ab4b5ab5b6ab6b7ab7b8ab8babb2ab2b3ab3b4ab4
ab4ab4b4ab5b5ab6b6ab7b7ab8b8abbab2b2ab3b3ab4b4
b5b5ab7b6ab8b7abb8ab2bab3b2ab4b3ab5b4ab6b5ab7
ab5ab5b7ab6b8ab7bab8b2abb3ab2b4ab3b5ab4b6ab5b7
b6b6ab2b7ab3b8ab4bab5b2ab6b3ab7b4ab8b5abb6ab2
ab6ab6b2ab7b3ab8b4abb5ab2b6ab3b7ab4b8ab5bab6b2
b7b7ab5b8ab6bab7b2ab8b3abb4ab2b5ab3b6ab4b7ab5
ab7ab7b5ab8b6abb7ab2b8ab3bab4b2ab5b3ab6b4ab7b5
b8b8ab8babb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
ab8ab8b8abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

18 unique, 90 total

Length:Presentation:Description:Related:
9 〈a, b | aa=1, abbbbb=b〉 Finite non-commutative monoid with 18 elements 22 isomorphic, 9 anti-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

14 total

Length:Presentation:
8〈a, b | aa=1, abbb=ba〉
8〈a, b | aa=1, bbb=aba〉
10〈a, b | aa=1, aaabbba=b〉
10〈a, b | aa=1, abaabba=b〉
10〈a, b | aa=1, aaabbb=ba〉
10〈a, b | aa=1, aabbba=ab〉
10〈a, b | aa=1, abaabb=ba〉
10〈a, b | aa=1, abbaab=ba〉
10〈a, b | aa=1, aaaba=bbb〉
10〈a, b | aa=1, aabbb=aba〉
10〈a, b | aa=1, abbba=aab〉
10〈a, b | aa=1, baabb=aba〉
10〈a, b | aa=1, abbb=aaba〉
10〈a, b | aa=1, baaa=abbb〉