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〈a, b | aab=ba, bbb=1〉

Monoid presentation of length 8

Properties

Completion parameters

Complete rewriting system

  1. a8 ⇒ a
  2. ab ⇒ ba4
  3. b3 ⇒ 1

Idempotents

2 elements

Cayley table

Idempotents are shown in bold.

1aba2bab2a3ba2b2aa4ba3b2a2a5ba4b2a3a6ba5b2a4a7ba6b2a5ba7b2a6b2a7
11aba2bab2a3ba2b2aa4ba3b2a2a5ba4b2a3a6ba5b2a4a7ba6b2a5ba7b2a6b2a7
aaa2ba4a3ba5b2a2a4ba6b2a3a5ba7b2a4a6bab2a5a7ba2b2a6aba3b2a7ba4b2ab2a2
bbbab2ba2b2a1ba3b2a2aba4b2a3a2ba5b2a4a3ba6b2a5a4ba7b2a6a5b2a7a6a7
a2a2a3baa4ba2b2a4a5ba3b2a5a6ba4b2a6a7ba5b2a7aba6b2aa2ba7b2a2bab2a3b2a4
bababa2b2a4ba3b2a5a2ba4b2a6a3ba5b2a7a4ba6b2aa5ba7b2a2a6bab2a3a7b2a4aa2
b2b2b2a1b2a2abb2a3a2bab2a4a3ba2b2a5a4ba3b2a6a5ba4b2a7a6ba5a7ba6ba7
a3a3a4ba5a5ba6b2a6a6ba7b2a7a7bab2aaba2b2a2a2ba3b2a3a3ba4b2a4ba5b2a5b2a6
ba2ba2ba3b2aba4b2a2a4ba5b2a3a5ba6b2a4a6ba7b2a5a7bab2a6aba2b2a7a2b2aa3a4
b2ab2ab2a2a4b2a3a5ba2b2a4a6ba3b2a5a7ba4b2a6aba5b2a7a2ba6b2aa3ba7a4baba2
a4a4a5ba2a6ba3b2aa7ba4b2a2aba5b2a3a2ba6b2a4a3ba7b2a5a4bab2a6ba2b2a7b2a
ba3ba3ba4b2a5ba5b2a6a6ba6b2a7a7ba7b2aabab2a2a2ba2b2a3a3ba3b2a4a4b2a5a5a6
b2a2b2a2b2a3ab2a4a2ba4b2a5a3ba5b2a6a4ba6b2a7a5ba7b2aa6bab2a2a7ba2aba3ba4
a5a5a6ba6a7ba7b2a3abab2a4a2ba2b2a5a3ba3b2a6a4ba4b2a7a5ba5b2aba6b2a2b2a3
ba4ba4ba5b2a2ba6b2a3aba7b2a4a2bab2a5a3ba2b2a6a4ba3b2a7a5ba4b2aa6b2a2a7a
b2a3b2a3b2a4a5b2a5a6ba6b2a6a7ba7b2a7abab2aa2ba2b2a2a3ba3b2a3a4ba4a5ba5ba6
a6a6a7ba3aba4b2a5a2ba5b2a6a3ba6b2a7a4ba7b2aa5bab2a2a6ba2b2a3ba3b2a4b2a5
ba5ba5ba6b2a6ba7b2a7a3bab2aa4ba2b2a2a5ba3b2a3a6ba4b2a4a7ba5b2a5ab2a6a2a3
b2a4b2a4b2a5a2b2a6a3bab2a7a4ba2b2aa5ba3b2a2a6ba4b2a3a7ba5b2a4aba6a2ba7ba
a7a7aba7a2bab2a7a3ba2b2aa4ba3b2a2a5ba4b2a3a6ba5b2a4a7ba6b2a5ba7b2a6b2a7
ba6ba6ba7b2a3bab2a4a5ba2b2a5a6ba3b2a6a7ba4b2a7aba5b2aa2ba6b2a2a3b2a3a4a5
b2a5b2a5b2a6a6b2a7a7ba3b2aaba4b2a2a2ba5b2a3a3ba6b2a4a4ba7b2a5a5baa6ba2ba3
ba7ba7bab2a7ba2b2aa7ba3b2a2aba4b2a3a2ba5b2a4a3ba6b2a5a4ba7b2a6a5b2a7a6a7
b2a6b2a6b2a7a3b2aa4ba5b2a2a5ba6b2a3a6ba7b2a4a7bab2a5aba2b2a6a2ba3a3ba4ba5
b2a7b2a7b2aa7b2a2aba7b2a3a2bab2a4a3ba2b2a5a4ba3b2a6a5ba4b2a7a6ba5a7ba6ba7

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

14 unique, 71 total

Length:Presentation:Description:Related:
9 〈a, b | aaa=1, aaba=bb〉 Finite non-commutative monoid with 24 elements 5 isomorphic
9 〈a, b | aaa=1, abab=ba〉 Finite non-commutative monoid with 24 elements 2 isomorphic, 3 anti-isomorphic
10 〈a, b | aba=bb, aabbb=1〉 Finite non-Abelian group with 24 elements 22 isomorphic
10 〈a, b | bb=aa, aaabab=1〉 Finite non-Abelian group with 24 elements 2 isomorphic
10 〈a, b | aaa=1, abaab=ba〉 Finite non-commutative monoid with 24 elements 1 isomorphic
11 〈a, b | ababa=b, abbaa=1〉 Finite non-Abelian group with 24 elements 2 isomorphic
11 〈a, b | aaaab=1, bbbbbb=1〉 Isomorphic to ℤ24 8 isomorphic
11 〈a, b | aaabb=1, bababa=1〉 Finite non-Abelian group with 24 elements 3 isomorphic
11 〈a, b | abba=b, aaabbb=1〉 Finite non-Abelian group with 24 elements 2 isomorphic
11 〈a, b | abba=b, ababab=1〉 Finite non-Abelian group with 24 elements
11 〈a, b | aba=b, aaabbbb=1〉 Finite non-Abelian group with 24 elements 7 isomorphic
11 〈a, b | aba=a, aaaab=bb〉 Finite non-commutative monoid with 24 elements
11 〈a, b | aba=b, aaaa=bab〉 Finite non-commutative monoid with 24 elements
11 〈a, b | aaa=1, aabaaba=b〉 Finite non-commutative monoid with 24 elements

Other isomorphic instances

6 total

Length:Presentation:
9〈a, b | aaa=1, aabba=b〉
10〈a, b | aaa=1, aabb=baa〉
10〈a, b | aaa=1, abba=aab〉
11〈a, b | aaa=1, aaabba=ab〉
11〈a, b | aaa=1, aaaab=bba〉
11〈a, b | aaa=1, aabaa=abb〉