#24186 ⟨a, b | aa=a, bbbbbbb=a⟩

Up:Monoid enumeration
Prev:#24184 ⟨a, b | aa=a, bbbabbb=a⟩
Next:#24187 ⟨a, b | aa=a, bbbbbbb=b⟩

Contents

  1. Properties
  2. Commutative structure
  3. Finite structure
  4. Staircase diagram
  5. Cayley table
  6. Right Cayley graph
  7. Rewriting system
  8. Same cardinality

Properties

Commutative structure

Finite structure

An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1bb2b3b4b5b6b7b8b9b10b11b12b13
11bb2b3b4b5b6b7b8b9b10b11b12b13
bbb2b3b4b5b6b7b8b9b10b11b12b13b7
b2b2b3b4b5b6b7b8b9b10b11b12b13b7b8
b3b3b4b5b6b7b8b9b10b11b12b13b7b8b9
b4b4b5b6b7b8b9b10b11b12b13b7b8b9b10
b5b5b6b7b8b9b10b11b12b13b7b8b9b10b11
b6b6b7b8b9b10b11b12b13b7b8b9b10b11b12
b7b7b8b9b10b11b12b13b7b8b9b10b11b12b13
b8b8b9b10b11b12b13b7b8b9b10b11b12b13b7
b9b9b10b11b12b13b7b8b9b10b11b12b13b7b8
b10b10b11b12b13b7b8b9b10b11b12b13b7b8b9
b11b11b12b13b7b8b9b10b11b12b13b7b8b9b10
b12b12b13b7b8b9b10b11b12b13b7b8b9b10b11
b13b13b7b8b9b10b11b12b13b7b8b9b10b11b12

Right Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:
Word:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. b14 ⇒ b7 [4]
2. a ⇒ b7 [2]
# ab:aa=a,bbbbbbb=a b/a - -
bbbbbbbbbbbbbb=bbbbbbb
a=bbbbbbb

Same cardinality

30 unique, 316 total

Σ#PresentationPropertiesDescriptionφ
81123⟨a, b | aa=1, abbbb=b⟩Noc 1459
91682⟨a, b | aba=bb, bab=a⟩Noc 141
93034⟨a, b | aa=a, bbbb=ab⟩Noc 142
103773⟨a, b | aaaa=bb, abbb=1⟩Grp Com 14ℤ14165
105218⟨a, b | aab=bb, bbba=a⟩Noc 146
105404⟨a, b | aab=bb, aba=aa⟩Noc 141
106718⟨a, b | aab=b, bbba=aa⟩Noc 142
1112183⟨a, b | aaaa=ab, baab=b⟩Noc 143
1112206⟨a, b | aaaa=bb, abbb=a⟩Noc Com 141
1112207⟨a, b | aaaa=bb, abbb=b⟩Noc Com 141
1112441⟨a, b | aabb=aa, baab=b⟩Noc 143
1112499⟨a, b | abab=aa, abba=b⟩Noc 14
1114383⟨a, b | aaaa=b, aabbb=a⟩Noc Com 14ℕ(14 = 1)15
1114384⟨a, b | aaaa=b, aabbb=b⟩Noc Com 14ℕ(14 = 4)5
1115532⟨a, b | aaa=bb, abbbb=b⟩Noc Com 14ℕ(14 = 3)11
1115539⟨a, b | aaa=bb, babbb=a⟩Noc Com 14
1116020⟨a, b | aaa=ab, baab=bb⟩Noc 141
1116079⟨a, b | aaa=bb, bbbb=ab⟩Noc 14
1116293⟨a, b | aab=bb, abab=aa⟩Noc 14
1116470⟨a, b | aba=bb, bbbb=aa⟩Noc 14
1119552⟨a, b | aab=b, abbaa=aa⟩Noc 142
1120844⟨a, b | ab=aa, bbbbb=aa⟩Noc 141
1120846⟨a, b | ab=aa, bbbbb=ba⟩Noc 14
1121023⟨a, b | ab=aa, bbaa=bbb⟩Noc 141
1121040⟨a, b | ab=aa, bbbb=aaa⟩Noc 143
1121044⟨a, b | ab=aa, bbbb=baa⟩Noc 141
1121046⟨a, b | ab=aa, bbbb=bba⟩Noc 14
1121110⟨a, b | bb=aa, abab=aaa⟩Noc 142
1124331⟨a, b | aa=b, bbbbbbb=b⟩Noc Com 14ℕ(14 = 2)
1125055⟨a, b | ab=a, bbaaaa=bb⟩Noc 14