#12206 ⟨a, b | aaaa=bb, abbb=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.

1aba2abb2a3a2bab2b3a3ba2b2b4a3b2
11aba2abb2a3a2bab2b3a3ba2b2b4a3b2
aaa2aba3a2bab2b2a3ba2b2ab3a3b2abb4
bbabb2a2bab2b3a3ba2b2ab4a3b2a2b2a3
a2a2a3a2bb2a3ba2b2ab2b3a3b2a2ab4a2bab
ababa2bab2a3ba2b2ab3a3b2a2abb4a3ab2b2
b2b2ab2b3a2b2ab4a3b2a2abb2a3a2bb3a3b
a3a3b2a3bab2b3a3b2a2b2ab4a3a2aba3ba2b
a2ba2ba3ba2b2b3a3b2a2ab4a3a2babb2a2b2ab2
ab2ab2a2b2aa3b2a2abb4a3a2bab2b2a3bab3
b3b3ab4a2abb2a3a2bab2b3a3ba2b2b4a3b2
a3ba3bb3a3b2ab4a3a2abb2a3ba2bab2a3b2a2b2
a2b2a2b2a3b2a2b4a3a2babb2a3ba2b2ab2b3a2a
b4b4abb2a2bab2b3a3ba2b2ab4a3b2a2b2a3
a3b2a3b2b4a3abb2a3ba2bab2b3a3b2a2b2aa3a2

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. ba ⇒ ab [7]
2. a4 ⇒ b2 [1]
3. ab3 ⇒ a [2]
4. b5 ⇒ b2 [4]
# ab:aaaa=bb,abbb=a ab - -
ba=ab
aaaa=bb
abbb=a
bbbbb=bb

Same cardinality

30 unique, 315 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
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.

1 total

Σ#PresentationMapping
1112210⟨a, b | aaaa=bb, babb=a⟩φ(a) = a, φ(b) = b