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〈a, b | aa=1, abba=b〉

Monoid presentation of length 7

Properties

Completion parameters

Complete rewriting system

  1. b4 ⇒ b
  2. ba ⇒ ab2
  3. a2 ⇒ 1

Idempotents

2 elements

Cayley table

Idempotents are shown in bold.

1ababb2ab2b3ab3
11ababb2ab2b3ab3
aa1abbab2b2ab3b3
bbab2b2ab3b3abbab2
ababb2ab2b3ab3babb2
b2b2abb3ab2bab3b2ab
ab2ab2bab3b2abb3ab2b
b3b3ab3babb2ab2b3ab3
ab3ab3b3abbab2b2ab3b3

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

44 unique, 1165 total

Length:Presentation:Description:Related:
7 〈a, b | aab=1, bbbb=1〉 Isomorphic to ℤ8 727 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, baaa=b〉 Finite non-commutative monoid with 8 elements 6 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

46 total

Length:Presentation:
7〈a, b | aa=1, aba=bb〉
7〈a, b | aa=1, abb=ba〉
9〈a, b | aa=1, aaabba=b〉
9〈a, b | aa=1, abaaba=b〉
9〈a, b | aa=1, aaaba=bb〉
9〈a, b | aa=1, aaabb=ba〉
9〈a, b | aa=1, aabba=ab〉
9〈a, b | aa=1, abaab=ba〉
9〈a, b | aa=1, aaab=bba〉
9〈a, b | aa=1, aaba=abb〉
9〈a, b | aa=1, aabb=aba〉
9〈a, b | aa=1, abba=aab〉
9〈a, b | aa=1, baab=aba〉
11〈a, b | aa=1, aaaaabba=b〉
11〈a, b | aa=1, aaabaaba=b〉
11〈a, b | aa=1, aaabbaaa=b〉
11〈a, b | aa=1, abaaaaba=b〉
11〈a, b | aa=1, aaaaaba=bb〉
11〈a, b | aa=1, aaaaabb=ba〉
11〈a, b | aa=1, aaaabba=ab〉
11〈a, b | aa=1, aaabaaa=bb〉
11〈a, b | aa=1, aaabaab=ba〉
11〈a, b | aa=1, aaabbaa=ba〉
11〈a, b | aa=1, aabaaba=ab〉
11〈a, b | aa=1, abaaaab=ba〉
11〈a, b | aa=1, aaaaab=bba〉
11〈a, b | aa=1, aaaaba=abb〉
11〈a, b | aa=1, aaaabb=aba〉
11〈a, b | aa=1, aaabaa=bba〉
11〈a, b | aa=1, aaabba=aab〉
11〈a, b | aa=1, aaabba=baa〉
11〈a, b | aa=1, aabaab=aba〉
11〈a, b | aa=1, aabbaa=aba〉
11〈a, b | aa=1, abaaba=aab〉
11〈a, b | aa=1, baaaab=aba〉
11〈a, b | aa=1, aaaab=abba〉
11〈a, b | aa=1, aaaba=aabb〉
11〈a, b | aa=1, aaaba=baab〉
11〈a, b | aa=1, aaaba=bbaa〉
11〈a, b | aa=1, aaabb=aaba〉
11〈a, b | aa=1, aaabb=baaa〉
11〈a, b | aa=1, aabaa=abba〉
11〈a, b | aa=1, aabba=aaab〉
11〈a, b | aa=1, aabba=abaa〉
11〈a, b | aa=1, abaab=aaba〉
11〈a, b | aa=1, abaab=baaa〉