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〈a, b | aab=a, bbb=1〉

Monoid presentation of length 7

Properties

Completion parameters

Complete rewriting system

  1. a4 ⇒ a
  2. ab ⇒ a3
  3. b3 ⇒ 1

Idempotents

4 elements

Cayley table

Idempotents are shown in bold.

1aba2bab2a3ba2b2aba3b2a2b2a3
11aba2bab2a3ba2b2aba3b2a2b2a3
aaa2a3a3aa2aa2a3a3aa2
bbbab2ba2b2a1ba3b2a2ab2a3a2a3
a2a2a3aaa2a3a2a3aaa2a3
bababa2ba3ba3baba2baba2ba3ba3baba2
b2b2b2a1b2a2abb2a3a2baa3ba2ba3
a3a3aa2a2a3aa3aa2a2a3a
ba2ba2ba3bababa2ba3ba2ba3bababa2ba3
b2ab2ab2a2b2a3b2a3b2ab2a2b2ab2a2b2a3b2a3b2ab2a2
ba3ba3baba2ba2ba3baba3baba2ba2ba3ba
b2a2b2a2b2a3b2ab2ab2a2b2a3b2a2b2a3b2ab2ab2a2b2a3
b2a3b2a3b2ab2a2b2a2b2a3b2ab2a3b2ab2a2b2a2b2a3b2a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

51 unique, 639 total

Length:Presentation:Description:Related:
7 〈a, b | ab=aa, bbb=1〉 Finite non-commutative monoid with 12 elements 9 isomorphic
8 〈a, b | aba=ab, bbb=1〉 Finite non-commutative monoid with 12 elements
8 〈a, b | aaaa=1, abbb=1〉 Isomorphic to ℤ12 325 isomorphic
8 〈a, b | ab=aa, bbb=b〉 Finite non-commutative monoid with 12 elements
8 〈a, b | aaa=1, aabb=b〉 Finite non-commutative monoid with 12 elements 16 isomorphic
9 〈a, b | aab=bb, aba=a〉 Finite non-commutative monoid with 12 elements 2 isomorphic
9 〈a, b | aba=b, aaabb=1〉 Finite non-Abelian group with 12 elements 19 isomorphic
9 〈a, b | aaa=a, bbbb=a〉 Finite commutative monoid with 12 elements 6 isomorphic
9 〈a, b | aaa=b, bbbb=a〉 Finite commutative monoid with 12 elements 27 isomorphic
9 〈a, b | aaa=b, bbbb=b〉 Finite commutative monoid with 12 elements 23 isomorphic
9 〈a, b | ab=aa, bbb=bb〉 Finite non-commutative monoid with 12 elements
9 〈a, b | aaa=1, aabab=b〉 Finite non-commutative monoid with 12 elements 8 isomorphic, 4 anti-isomorphic
10 〈a, b | bab=aba, aabb=1〉 Finite non-Abelian group with 12 elements 4 isomorphic
10 〈a, b | aaa=aa, bbbb=a〉 Finite commutative monoid with 12 elements
10 〈a, b | aaa=ab, bbbb=a〉 Finite commutative monoid with 12 elements 7 isomorphic
10 〈a, b | aaa=bb, bbbb=a〉 Finite commutative monoid with 12 elements 43 isomorphic
10 〈a, b | aab=bb, abba=a〉 Finite non-commutative monoid with 12 elements 6 isomorphic
10 〈a, b | aaa=ab, bab=bb〉 Finite non-commutative monoid with 12 elements 1 isomorphic
10 〈a, b | aaa=a, aaab=bb〉 Finite non-commutative monoid with 12 elements
10 〈a, b | aaa=b, bbbb=bb〉 Finite commutative monoid with 12 elements 4 isomorphic
10 〈a, b | aab=a, bbbb=ba〉 Finite non-commutative monoid with 12 elements
10 〈a, b | aab=a, bbbb=bb〉 Finite non-commutative monoid with 12 elements
10 〈a, b | aba=a, bbbb=ab〉 Finite non-commutative monoid with 12 elements
10 〈a, b | ab=aa, bbaa=bb〉 Finite non-commutative monoid with 12 elements 2 isomorphic
11 〈a, b | aaaa=aa, abab=b〉 Finite non-commutative monoid with 12 elements
11 〈a, b | aaab=bb, aaba=a〉 Finite non-commutative monoid with 12 elements 2 isomorphic
11 〈a, b | aab=aaa, baaa=b〉 Finite non-commutative monoid with 12 elements 4 isomorphic
11 〈a, b | aab=aaa, bbbb=a〉 Finite commutative monoid with 12 elements 2 isomorphic
11 〈a, b | aab=ba, ababbb=1〉 Finite non-Abelian group with 12 elements 5 isomorphic
11 〈a, b | aaa=aa, abbbb=b〉 Finite non-commutative monoid with 12 elements
11 〈a, b | aaa=ab, aabb=bb〉 Finite non-commutative monoid with 12 elements 2 isomorphic
11 〈a, b | aaa=bb, aabb=ab〉 Finite non-commutative monoid with 12 elements 2 isomorphic
11 〈a, b | aaa=bb, aabb=ba〉 Finite non-commutative monoid with 12 elements 1 isomorphic
11 〈a, b | aab=aa, abab=bb〉 Finite non-commutative monoid with 12 elements
11 〈a, b | aab=bb, aaaa=ab〉 Finite non-commutative monoid with 12 elements 1 isomorphic
11 〈a, b | aab=bb, aaaa=ba〉 Finite non-commutative monoid with 12 elements
11 〈a, b | aba=bb, aabb=aa〉 Finite non-commutative monoid with 12 elements 2 isomorphic
11 〈a, b | aab=a, baaaa=bb〉 Finite non-commutative monoid with 12 elements
11 〈a, b | aba=b, bbbbb=aa〉 Finite commutative monoid with 12 elements
11 〈a, b | aaa=b, bbbb=abb〉 Finite commutative monoid with 12 elements 2 isomorphic
11 〈a, b | aab=a, bbaa=bbb〉 Finite non-commutative monoid with 12 elements
11 〈a, b | aab=b, aaaa=bba〉 Finite non-commutative monoid with 12 elements
11 〈a, b | ab=aa, baaaaa=b〉 Finite non-commutative monoid with 12 elements 30 isomorphic
11 〈a, b | bb=aa, aaaaab=a〉 Finite commutative monoid with 12 elements 15 isomorphic
11 〈a, b | ab=aa, baaaa=bb〉 Finite non-commutative monoid with 12 elements 7 isomorphic
11 〈a, b | bb=aa, aaaa=aba〉 Finite non-commutative monoid with 12 elements 4 isomorphic
11 〈a, b | bb=aa, aaab=aba〉 Finite non-commutative monoid with 12 elements 3 isomorphic
11 〈a, b | bb=aa, abab=aba〉 Finite non-commutative monoid with 12 elements
11 〈a, b | aa=a, abbbbbb=b〉 Finite non-commutative monoid with 12 elements
11 〈a, b | ab=a, baaaaa=bb〉 Finite non-commutative monoid with 12 elements
11 〈a, b | ab=a, bbaaa=bbb〉 Finite non-commutative monoid with 12 elements

Other isomorphic instances

8 total

Length:Presentation:
8〈a, b | abb=aa, bbb=1〉
9〈a, b | aaa=1, abba=ab〉
10〈a, b | aaa=1, aaabba=b〉
10〈a, b | aaa=1, aaab=bba〉
11〈a, b | aaa=1, aaabaa=bb〉
11〈a, b | aaa=1, aaabb=baa〉
11〈a, b | aaa=1, aabba=aab〉
11〈a, b | aaa=1, abaab=aba〉

Other anti-isomorphic instances

12 total

Length:Presentation:
9〈a, b | aaa=1, aabb=ab〉
9〈a, b | aaa=1, baab=ab〉
10〈a, b | aaa=1, aaaabb=b〉
10〈a, b | aaa=1, aabaab=b〉
10〈a, b | aaa=1, abaaab=b〉
10〈a, b | aaa=1, aaab=abb〉
10〈a, b | aaa=1, aaba=bba〉
11〈a, b | aaa=1, aaaaab=bb〉
11〈a, b | aaa=1, aaabb=aab〉
11〈a, b | aaa=1, aabba=aba〉
11〈a, b | aaa=1, abaab=aab〉
11〈a, b | aaa=1, baaab=aab〉