#1603 ⟨a, b | aaa=bb, aba=b⟩

Up:Monoid enumeration
Prev:#1602 ⟨a, b | aaa=bb, aba=a⟩
Next:#1604 ⟨a, b | aaa=bb, abb=a⟩

Contents

  1. Properties
  2. Finite structure
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Finite structure

Elements in the center commute with all other elements.
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.

Cayley table

Idempotents are shown in bold.

1aba2baa3ba2a4ba3a5ba4a6ba5a7a8
11aba2baa3ba2a4ba3a5ba4a6ba5a7a8
aaa2ba5a3ba4baa5ba2a6ba3a7ba4a8a3
bbbaa3ba2a4ba3a5ba4a6ba5a7ba8baba2
a2a2a3ba4a4ba5a5ba6baa7ba2a8ba3a3a4
bababa2a8ba3a3ba4a4ba5a5ba6baa7ba2ba3
a3a3a4ba3a5ba4a6ba5a7ba8baa3ba2a4a5
ba2ba2ba3a7ba4a8ba5a3ba4baa5ba2a6ba3ba4
a4a4a5ba2a6ba3a7ba4a8ba5a3ba4baa5a6
ba3ba3ba4a6ba5a7ba8baa3ba2a4ba3a5ba4ba5
a5a5a6baa7ba2a8ba3a3ba4a4ba5a5ba6a7
ba4ba4ba5a5ba6baa7ba2a8ba3a3ba4a4ba5b
a6a6a7ba8baa3ba2a4ba3a5ba4a6ba5a7a8
ba5ba5ba4baa5ba2a6ba3a7ba4a8ba5a3bba
a7a7a8ba5a3ba4baa5ba2a6ba3a7ba4a8a3
a8a8a3ba4a4ba5a5ba6baa7ba2a8ba3a3a4

Right Cayley graph

Idempotents are shown in bold.

Left 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. a9 ⇒ a3 [7]
2. ba6 ⇒ b [6]
3. ab ⇒ ba5 [5]
4. b2 ⇒ a3 [1]
# ab:aaa=bb,aba=b a/b - -
aaaaaaaaa=aaa
baaaaaa=b
ab=baaaaa
bb=aaa

Same cardinality

22 unique, 189 total

Σ#PresentationPropertiesDescriptionφ
91602⟨a, b | aaa=bb, aba=a⟩Noc 152
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)
1115503⟨a, b | aaa=ab, bbbbb=a⟩Noc Com 15ℕ(15 = 6)1
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
1120278⟨a, b | aba=b, abab=aaa⟩φ(a) = a, φ(b) = b