#3034 ⟨a, b | aa=a, bbbb=ab⟩

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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.
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.

1ababbab2abaab2b2ab3ab2aab3b3aab3a
11ababbab2abaab2b2ab3ab2aab3b3aab3a
aaaabababaab2abaab2ab2aab3ab2aab3ab3aab3a
bbbab2ab2b2ab3ab2aab3b3aabab3aababaaba
abababaab2ab2ab2aab3ab2aab3ab3aabab3aababaaba
bababaab2ab2ab2aab3ab2aab3ab3aabab3aababaaba
b2b2b2ab3ab3b3aabab3aababaab2abaab2ab2aab2a
abaabaabaab2ab2ab2aab3ab2aab3ab3aabab3aababaaba
ab2ab2ab2aab3ab3ab3aabab3aababaab2abaab2ab2aab2a
b2ab2ab2aab3ab3ab3aabab3aababaab2abaab2ab2aab2a
b3b3b3aabababaab2abaab2ab2aab3ab2aab3ab3aab3a
ab2aab2aab2aab3ab3ab3aabab3aababaab2abaab2ab2aab2a
ab3ab3ab3aabababaab2abaab2ab2aab3ab2aab3ab3aab3a
b3ab3ab3aabababaab2abaab2ab2aab3ab2aab3ab3aab3a
ab3aab3aab3aabababaab2abaab2ab2aab3ab2aab3ab3aab3a

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. a2 ⇒ a [1]
2. bab ⇒ ab2 [3]
3. b4 ⇒ ab [2]
# ab:aa=a,bbbb=ab ab - -
aa=a
bab=abb
bbbb=ab

Same cardinality

30 unique, 314 total

Σ#PresentationPropertiesDescriptionφ
81123⟨a, b | aa=1, abbbb=b⟩Noc 1459
91682⟨a, b | aba=bb, bab=a⟩Noc 141
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
1124186⟨a, b | aa=a, bbbbbbb=a⟩Noc Com 14ℕ(14 = 7)
1124331⟨a, b | aa=b, bbbbbbb=b⟩Noc Com 14ℕ(14 = 2)
1125055⟨a, b | ab=a, bbaaaa=bb⟩Noc 14

Isomorphic instances

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

2 total

Σ#PresentationMapping
109179⟨a, b | aa=a, bbbb=aab⟩φ(a) = a, φ(b) = b
1125718⟨a, b | aa=a, bbbb=aaab⟩φ(a) = a, φ(b) = b