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〈a, b | ab=aa, baaa=b〉

Monoid presentation of length 9

Properties

Completion parameters

Complete rewriting system

  1. ba ⇒ b2
  2. a2 ⇒ ab
  3. b4 ⇒ b

Idempotents

3 elements

Cayley table

Idempotents are shown in bold.

1ababb2ab2b3ab3
11ababb2ab2b3ab3
aaababab2ab2ab3ab3ab
bbb2b2b3b3bbb2
ababab2ab2ab3ab3ababab2
b2b2b3b3bbb2b2b3
ab2ab2ab3ab3ababab2ab2ab3
b3b3bbb2b2b3b3b
ab3ab3ababab2ab2ab3ab3ab

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

44 unique, 1205 total

Length:Presentation:Description:Related:
7 〈a, b | aab=1, bbbb=1〉 Isomorphic to ℤ8 727 isomorphic
7 〈a, b | aa=1, abba=b〉 Finite non-commutative monoid with 8 elements 46 isomorphic
8 〈a, b | aba=b, aabb=1〉 Finite non-Abelian group with 8 elements 28 isomorphic
8 〈a, b | aa=a, bbbb=a〉 Finite commutative monoid with 8 elements 48 isomorphic
8 〈a, b | aa=b, bbbb=a〉 Finite commutative monoid with 8 elements 34 isomorphic
8 〈a, b | aa=b, bbbb=b〉 Finite commutative monoid with 8 elements 46 isomorphic
8 〈a, b | aa=a, aba=bb〉 Finite non-commutative monoid with 8 elements 5 isomorphic
8 〈a, b | aa=1, ababa=b〉 Finite non-commutative monoid with 8 elements 13 isomorphic
9 〈a, b | aaa=bb, abb=b〉 Finite commutative monoid with 8 elements 55 isomorphic
9 〈a, b | aaa=bb, bab=a〉 Finite commutative monoid with 8 elements 9 isomorphic
9 〈a, b | aab=aa, baa=b〉 Finite non-commutative monoid with 8 elements 8 isomorphic
9 〈a, b | aab=bb, baa=a〉 Finite non-commutative monoid with 8 elements 4 anti-isomorphic
9 〈a, b | ab=aa, bbbb=a〉 Finite commutative monoid with 8 elements 32 isomorphic
9 〈a, b | bb=aa, aaab=a〉 Finite commutative monoid with 8 elements 3 isomorphic
9 〈a, b | ab=aa, baa=bb〉 Finite non-commutative monoid with 8 elements 1 isomorphic
9 〈a, b | ab=aa, bbb=aa〉 Finite non-commutative monoid with 8 elements 1 isomorphic
9 〈a, b | ab=aa, bbb=ba〉 Finite non-commutative monoid with 8 elements
9 〈a, b | aa=a, abbbb=b〉 Finite non-commutative monoid with 8 elements 14 isomorphic
9 〈a, b | ab=a, baaa=bb〉 Finite non-commutative monoid with 8 elements 9 isomorphic
9 〈a, b | ab=a, bbaa=bb〉 Finite non-commutative monoid with 8 elements 9 isomorphic
10 〈a, b | aab=bb, aaaa=b〉 Finite commutative monoid with 8 elements 18 isomorphic
10 〈a, b | aab=bb, abbb=a〉 Finite commutative monoid with 8 elements 10 isomorphic
10 〈a, b | aba=aa, aaab=b〉 Finite non-commutative monoid with 8 elements 3 isomorphic
10 〈a, b | aab=a, bbaa=bb〉 Finite non-commutative monoid with 8 elements
10 〈a, b | aab=b, aaaa=ba〉 Finite non-commutative monoid with 8 elements
10 〈a, b | aab=b, abaa=aa〉 Finite non-commutative monoid with 8 elements 1 isomorphic
10 〈a, b | aab=b, abbb=aa〉 Finite commutative monoid with 8 elements 4 isomorphic
10 〈a, b | ab=aa, aaaa=bb〉 Finite non-commutative monoid with 8 elements 7 isomorphic
10 〈a, b | ab=a, bbbb=baa〉 Finite non-commutative monoid with 8 elements 2 isomorphic
11 〈a, b | aabb=aa, abaa=b〉 Finite commutative monoid with 8 elements 7 isomorphic
11 〈a, b | aabb=aa, bbba=b〉 Finite non-commutative monoid with 8 elements
11 〈a, b | abbb=aa, bbbb=a〉 Finite commutative monoid with 8 elements 14 isomorphic
11 〈a, b | aab=b, aabaa=aa〉 Finite non-commutative monoid with 8 elements 2 isomorphic
11 〈a, b | aab=a, bbbb=bbb〉 Finite non-commutative monoid with 8 elements
11 〈a, b | aab=b, aaaa=aba〉 Finite non-commutative monoid with 8 elements
11 〈a, b | aab=b, aaaa=abb〉 Finite commutative monoid with 8 elements 1 isomorphic
11 〈a, b | ab=a, aaaaaa=bb〉 Finite commutative monoid with 8 elements
11 〈a, b | ab=a, bbbbbb=aa〉 Finite commutative monoid with 8 elements
11 〈a, b | ab=a, bbbbbb=ba〉 Finite non-commutative monoid with 8 elements
11 〈a, b | ab=a, aaaaa=bbb〉 Finite commutative monoid with 8 elements
11 〈a, b | ab=a, bbbbb=aaa〉 Finite commutative monoid with 8 elements
11 〈a, b | ab=a, bbbbb=bba〉 Finite non-commutative monoid with 8 elements
11 〈a, b | ab=a, bbbb=aaaa〉 Finite commutative monoid with 8 elements
11 〈a, b | ab=a, bbbb=bbba〉 Finite non-commutative monoid with 8 elements

Other isomorphic instances

6 total

Length:Presentation:
9〈a, b | ab=aa, baab=b〉
9〈a, b | ab=aa, baba=b〉
9〈a, b | ab=aa, babb=b〉
9〈a, b | ab=aa, bbaa=b〉
9〈a, b | ab=aa, bbab=b〉
9〈a, b | ab=aa, bbba=b〉