#15503 ⟨a, b | aaa=ab, bbbbb=a⟩

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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
  9. Isomorphic instances

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.

1bb2b3b4b5b6b7b8b9b10b11b12b13b14
11bb2b3b4b5b6b7b8b9b10b11b12b13b14
bbb2b3b4b5b6b7b8b9b10b11b12b13b14b6
b2b2b3b4b5b6b7b8b9b10b11b12b13b14b6b7
b3b3b4b5b6b7b8b9b10b11b12b13b14b6b7b8
b4b4b5b6b7b8b9b10b11b12b13b14b6b7b8b9
b5b5b6b7b8b9b10b11b12b13b14b6b7b8b9b10
b6b6b7b8b9b10b11b12b13b14b6b7b8b9b10b11
b7b7b8b9b10b11b12b13b14b6b7b8b9b10b11b12
b8b8b9b10b11b12b13b14b6b7b8b9b10b11b12b13
b9b9b10b11b12b13b14b6b7b8b9b10b11b12b13b14
b10b10b11b12b13b14b6b7b8b9b10b11b12b13b14b6
b11b11b12b13b14b6b7b8b9b10b11b12b13b14b6b7
b12b12b13b14b6b7b8b9b10b11b12b13b14b6b7b8
b13b13b14b6b7b8b9b10b11b12b13b14b6b7b8b9
b14b14b6b7b8b9b10b11b12b13b14b6b7b8b9b10

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. b15 ⇒ b6 [4]
2. a ⇒ b5 [2]
# ab:aaa=ab,bbbbb=a b/a - -
bbbbbbbbbbbbbbb=bbbbbb
a=bbbbb

Same cardinality

22 unique, 189 total

Σ#PresentationPropertiesDescriptionφ
91602⟨a, b | aaa=bb, aba=a⟩Noc 152
91603⟨a, b | aaa=bb, aba=b⟩Noc 151
91751⟨a, b | aaab=1, bbbbb=1⟩Grp Com 15ℤ15138
91993⟨a, b | aaa=a, abbb=b⟩Noc 154
105342⟨a, b | aaa=bb, aba=aa⟩Noc 151
105344⟨a, b | aaa=bb, aba=bb⟩Noc 152
106276⟨a, b | aaa=a, bbbbb=a⟩Noc Com 15ℕ(15 = 5)
106316⟨a, b | aaa=b, bbbbb=a⟩Noc Com 15ℕ(15 = 1)
106317⟨a, b | aaa=b, bbbbb=b⟩Noc Com 15ℕ(15 = 3)
1112981⟨a, b | abb=aaa, baaa=b⟩Noc 153
1112985⟨a, b | abb=aaa, baba=b⟩Noc 157
1113123⟨a, b | aab=aaa, aba=bb⟩Noc 15
1113191⟨a, b | abb=aaa, bbb=ab⟩Noc 151
1113192⟨a, b | abb=aaa, bbb=ba⟩Noc 15
1115439⟨a, b | aaa=aa, bbbbb=a⟩Noc Com 15ℕ(15 = 10)
1115543⟨a, b | aaa=bb, bbbbb=a⟩Noc Com 15ℕ(15 = 2)1
1116041⟨a, b | aaa=ab, bbbb=aa⟩Noc 15
1116142⟨a, b | aab=aa, bbbb=ab⟩Noc 15
1119374⟨a, b | aaa=b, bbbbb=ab⟩Noc Com 15ℕ(15 = 4)
1119502⟨a, b | aab=a, bbbbb=ba⟩Noc 151
1120144⟨a, b | aab=b, bbaa=aaa⟩Noc 15
1120819⟨a, b | ab=aa, bbaaa=bb⟩Noc 156

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

1 total

Σ#PresentationMapping
1119375⟨a, b | aaa=b, bbbbb=bb⟩φ(a) = b, φ(b) = bbb