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〈a, b | aba=b, bab=a〉

Monoid presentation of length 8

Properties

Completion parameters

Complete rewriting system

  1. a5 ⇒ a
  2. ba4 ⇒ b
  3. ab ⇒ ba3
  4. b2 ⇒ a2

Idempotents

2 elements

Cayley table

Idempotents are shown in bold.

1aba2baa3ba2a4ba3
11aba2baa3ba2a4ba3
aaa2ba3a3ba4baaba2
bbbaa2ba2a3ba3a4ba
a2a2a3ba2a4ba3aba2ba
bababa2aba3a2ba3baa4
a3a3a4baaba2a2ba3a3b
ba2ba2ba3a4babaa2ba2a3
a4a4aba2baa3ba2a4ba3
ba3ba3ba3baa4ba2aba3a2

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

38 unique, 939 total

Length:Presentation:Description:Related:
7 〈a, b | aaa=b, bbb=1〉 Isomorphic to ℤ9 562 isomorphic
8 〈a, b | aaa=a, abb=b〉 Finite non-commutative monoid with 9 elements 2 isomorphic
8 〈a, b | aaa=a, bbb=a〉 Finite commutative monoid with 9 elements 48 isomorphic
8 〈a, b | aaa=b, bbb=a〉 Finite commutative monoid with 9 elements 62 isomorphic
9 〈a, b | aaa=aa, bbb=a〉 Finite commutative monoid with 9 elements 16 isomorphic
9 〈a, b | aaa=ab, bab=b〉 Finite non-commutative monoid with 9 elements 4 isomorphic
9 〈a, b | aaa=ab, bbb=a〉 Finite commutative monoid with 9 elements 40 isomorphic
9 〈a, b | aaa=bb, bbb=a〉 Finite commutative monoid with 9 elements 70 isomorphic
9 〈a, b | aaa=a, abab=b〉 Finite non-commutative monoid with 9 elements 2 isomorphic
9 〈a, b | aaa=a, abba=b〉 Finite non-commutative monoid with 9 elements 2 isomorphic
9 〈a, b | aab=b, abba=a〉 Finite non-commutative monoid with 9 elements 13 isomorphic
9 〈a, b | ab=aa, bba=bb〉 Finite non-commutative monoid with 9 elements
9 〈a, b | ab=a, baaaa=b〉 Finite non-commutative monoid with 9 elements 20 isomorphic
10 〈a, b | aab=aaa, baa=b〉 Finite non-commutative monoid with 9 elements 1 isomorphic
10 〈a, b | aab=aaa, bbb=a〉 Finite commutative monoid with 9 elements 1 isomorphic
10 〈a, b | abb=aaa, bbb=a〉 Finite commutative monoid with 9 elements 12 isomorphic
10 〈a, b | aaa=aa, abbb=b〉 Finite non-commutative monoid with 9 elements 3 isomorphic
10 〈a, b | aaa=bb, aaba=b〉 Finite commutative monoid with 9 elements
10 〈a, b | aab=aa, bbbb=a〉 Finite commutative monoid with 9 elements 5 isomorphic
10 〈a, b | aaa=bb, aab=ab〉 Finite non-commutative monoid with 9 elements 1 isomorphic
10 〈a, b | aaa=a, ababa=b〉 Finite non-commutative monoid with 9 elements
10 〈a, b | aab=a, baaaa=b〉 Finite non-commutative monoid with 9 elements
10 〈a, b | aba=a, aaaab=b〉 Finite non-commutative monoid with 9 elements
10 〈a, b | aab=b, bbb=aaa〉 Finite commutative monoid with 9 elements 1 isomorphic
10 〈a, b | bb=aa, aaaa=ab〉 Finite non-commutative monoid with 9 elements 2 isomorphic
10 〈a, b | bb=aa, aabb=ba〉 Finite non-commutative monoid with 9 elements
10 〈a, b | ab=a, baaa=bbb〉 Finite non-commutative monoid with 9 elements 3 isomorphic
10 〈a, b | ab=a, bbaa=bbb〉 Finite non-commutative monoid with 9 elements 2 isomorphic
11 〈a, b | aba=aaa, aaab=b〉 Finite non-commutative monoid with 9 elements 3 isomorphic
11 〈a, b | abb=aaa, aaab=b〉 Finite commutative monoid with 9 elements 7 isomorphic
11 〈a, b | aaab=a, baaaa=b〉 Finite non-commutative monoid with 9 elements 1 isomorphic
11 〈a, b | aab=b, aaaaa=ba〉 Finite non-commutative monoid with 9 elements
11 〈a, b | aab=b, aaaaa=bb〉 Finite commutative monoid with 9 elements
11 〈a, b | aab=b, abaa=aaa〉 Finite non-commutative monoid with 9 elements
11 〈a, b | aab=b, babb=aaa〉 Finite commutative monoid with 9 elements 3 isomorphic
11 〈a, b | aba=a, abab=bbb〉 Finite non-commutative monoid with 9 elements
11 〈a, b | ab=aa, aaaaa=bb〉 Finite non-commutative monoid with 9 elements 15 isomorphic
11 〈a, b | ab=a, bbbbb=baa〉 Finite non-commutative monoid with 9 elements

Other isomorphic instances

3 total

Length:Presentation:
10〈a, b | aba=b, aaabb=a〉
10〈a, b | aba=b, aabba=a〉
10〈a, b | aba=b, abaab=a〉