#5344 ⟨a, b | aaa=bb, aba=bb⟩

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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.
The zero element z satisfies zy = yz = z for all y.
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.

1aba2abbaa3a2bba2baba4ba3ba2ba5a6
11aba2abbaa3a2bba2baba4ba3ba2ba5a6
aaa2aba3a2ba3a4ba3a4ba3a5a5a5a6a6
bbbaa3ba2baba4ba3ba2ba5a5a5a6a6a6a6
a2a2a3a2ba4ba3a4a5a5a5a5a6a6a6a6a6
ababa3a4a4ba3a5a5a5a6a6a6a6a6a6a6
bababa2babba3ba2bba3a5a6a5a6a6a6a6a6a6
a3a3a4ba3a5a5a5a6a6a6a6a6a6a6a6a6
a2ba2ba4a5a5a5a6a6a6a6a6a6a6a6a6a6
ba2ba2ba3ba2ba5a6a5a6a6a6a6a6a6a6a6a6
babbabba3a5a5a6a6a6a6a6a6a6a6a6a6a6
a4a4a5a5a6a6a6a6a6a6a6a6a6a6a6a6
ba3ba3a5a6a6a6a6a6a6a6a6a6a6a6a6a6
ba2bba2ba5a6a6a6a6a6a6a6a6a6a6a6a6a6
a5a5a6a6a6a6a6a6a6a6a6a6a6a6a6a6
a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6

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. a7 ⇒ a6 [5]
2. ba4 ⇒ a5 [4]
3. aba ⇒ a3 [2]
4. a3b ⇒ ba3 [3]
5. b2 ⇒ a3 [1]
# ab:aaa=bb,aba=bb a/b - -
aaaaaaa=aaaaaa
baaaa=aaaaa
aba=aaa
aaab=baaa
bb=aaa

Same cardinality

22 unique, 188 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
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.

2 total

Σ#PresentationMapping
1116486⟨a, b | aaa=bb, aba=aaa⟩φ(a) = a, φ(b) = b
1116557⟨a, b | bab=aa, bbb=bab⟩φ(a) = b, φ(b) = a