#15532 ⟨a, b | aaa=bb, abbbb=b⟩

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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.

1aa2a3a4a5a6a7a8a9a10a11a12a13
11aa2a3a4a5a6a7a8a9a10a11a12a13
aaa2a3a4a5a6a7a8a9a10a11a12a13a3
a2a2a3a4a5a6a7a8a9a10a11a12a13a3a4
a3a3a4a5a6a7a8a9a10a11a12a13a3a4a5
a4a4a5a6a7a8a9a10a11a12a13a3a4a5a6
a5a5a6a7a8a9a10a11a12a13a3a4a5a6a7
a6a6a7a8a9a10a11a12a13a3a4a5a6a7a8
a7a7a8a9a10a11a12a13a3a4a5a6a7a8a9
a8a8a9a10a11a12a13a3a4a5a6a7a8a9a10
a9a9a10a11a12a13a3a4a5a6a7a8a9a10a11
a10a10a11a12a13a3a4a5a6a7a8a9a10a11a12
a11a11a12a13a3a4a5a6a7a8a9a10a11a12a13
a12a12a13a3a4a5a6a7a8a9a10a11a12a13a3
a13a13a3a4a5a6a7a8a9a10a11a12a13a3a4

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. a14 ⇒ a3 [3]
2. b ⇒ a7 [2]
# ab:aaa=bb,abbbb=b a/b - -
aaaaaaaaaaaaaa=aaa
b=aaaaaaa

Same cardinality

30 unique, 305 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
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
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.

11 total

Σ#PresentationMapping
1115540⟨a, b | aaa=bb, babbb=b⟩φ(a) = a, φ(b) = aaaaaaa
1115542⟨a, b | aaa=bb, bbabb=b⟩φ(a) = a, φ(b) = aaaaaaa
1118789⟨a, b | aaa=b, aabbbb=b⟩φ(a) = a, φ(b) = aaa
1118801⟨a, b | aaa=b, ababbb=b⟩φ(a) = a, φ(b) = aaa
1118805⟨a, b | aaa=b, abbabb=b⟩φ(a) = a, φ(b) = aaa
1118807⟨a, b | aaa=b, abbbab=b⟩φ(a) = a, φ(b) = aaa
1118809⟨a, b | aaa=b, abbbba=b⟩φ(a) = a, φ(b) = aaa
1118819⟨a, b | aaa=b, baabbb=b⟩φ(a) = a, φ(b) = aaa
1118821⟨a, b | aaa=b, bababb=b⟩φ(a) = a, φ(b) = aaa
1118823⟨a, b | aaa=b, babbab=b⟩φ(a) = a, φ(b) = aaa
1118827⟨a, b | aaa=b, bbaabb=b⟩φ(a) = a, φ(b) = aaa