Back

〈a, b | ab=aa, bb=1〉

Monoid presentation of length 6

Properties

Completion parameters

Complete rewriting system

  1. ab ⇒ a2
  2. b2 ⇒ 1
  3. a3 ⇒ a

Idempotents

3 elements

Cayley table

Idempotents are shown in bold.

1aba2baba2
11aba2baba2
aaa2a2aaa2
bbba1ba2aa2
a2a2aaa2a2a
bababa2ba2bababa2
ba2ba2bababa2ba2ba

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

25 unique, 1903 total

Length:Presentation:Description:Related:
6 〈a, b | aaa=1, abb=1〉 Isomorphic to ℤ6 1373 isomorphic
6 〈a, b | aa=1, abb=b〉 Finite non-commutative monoid with 6 elements 30 isomorphic
7 〈a, b | ab=aa, bb=b〉 Finite non-commutative monoid with 6 elements
7 〈a, b | aa=a, bbb=a〉 Finite commutative monoid with 6 elements 49 isomorphic
7 〈a, b | aa=b, bbb=a〉 Finite commutative monoid with 6 elements 61 isomorphic
7 〈a, b | aa=b, bbb=b〉 Finite commutative monoid with 6 elements 55 isomorphic
7 〈a, b | aa=1, abab=b〉 Finite non-commutative monoid with 6 elements 31 isomorphic, 22 anti-isomorphic
8 〈a, b | ab=aa, baa=b〉 Finite non-commutative monoid with 6 elements 4 isomorphic
8 〈a, b | ab=aa, bbb=a〉 Finite commutative monoid with 6 elements 46 isomorphic
8 〈a, b | aa=a, abbb=b〉 Finite non-commutative monoid with 6 elements 19 isomorphic
8 〈a, b | aa=a, aab=bb〉 Finite non-commutative monoid with 6 elements 3 isomorphic
8 〈a, b | ab=a, baa=bb〉 Finite non-commutative monoid with 6 elements 12 isomorphic, 1 anti-isomorphic
9 〈a, b | abba=b, baba=1〉 Finite non-Abelian group with 6 elements 66 isomorphic
9 〈a, b | aaa=aa, abb=b〉 Finite non-commutative monoid with 6 elements 5 isomorphic
9 〈a, b | aab=bb, abb=a〉 Finite commutative monoid with 6 elements 13 isomorphic
9 〈a, b | aba=aa, baa=b〉 Finite non-commutative monoid with 6 elements
9 〈a, b | abb=aa, bbb=a〉 Finite commutative monoid with 6 elements 49 isomorphic
9 〈a, b | ab=a, aaaa=bb〉 Finite commutative monoid with 6 elements 14 isomorphic
9 〈a, b | ab=a, bbbb=aa〉 Finite commutative monoid with 6 elements 5 isomorphic
9 〈a, b | ab=a, bbbb=ba〉 Finite non-commutative monoid with 6 elements 2 isomorphic
9 〈a, b | ab=a, bbb=aaa〉 Finite commutative monoid with 6 elements 9 isomorphic
9 〈a, b | ab=a, bbb=bba〉 Finite non-commutative monoid with 6 elements 3 isomorphic
10 〈a, b | aba=aa, aabb=b〉 Finite non-commutative monoid with 6 elements 5 isomorphic
10 〈a, b | aab=b, aaba=aa〉 Finite non-commutative monoid with 6 elements 1 isomorphic
11 〈a, b | aa=a, abbbbba=b〉 Finite commutative monoid with 6 elements

Other isomorphic instances

22 total

Length:Presentation:
8〈a, b | aa=1, aabba=b〉
8〈a, b | aa=1, aaba=bb〉
8〈a, b | aa=1, aabb=ba〉
8〈a, b | aa=1, abba=ab〉
8〈a, b | aa=1, abb=aba〉
10〈a, b | aa=1, aaaabba=b〉
10〈a, b | aa=1, aabaaba=b〉
10〈a, b | aa=1, aaaaba=bb〉
10〈a, b | aa=1, aaaabb=ba〉
10〈a, b | aa=1, aaabba=ab〉
10〈a, b | aa=1, aabaab=ba〉
10〈a, b | aa=1, abaaba=ab〉
10〈a, b | aa=1, aaaab=bba〉
10〈a, b | aa=1, aaaba=abb〉
10〈a, b | aa=1, aaabb=aba〉
10〈a, b | aa=1, aabba=aab〉
10〈a, b | aa=1, aabba=baa〉
10〈a, b | aa=1, abaab=aba〉
10〈a, b | aa=1, aabb=aaba〉
10〈a, b | aa=1, abba=aaab〉
10〈a, b | aa=1, baaa=aabb〉
10〈a, b | aa=1, baab=aaba〉