#6293 ⟨a, b | aaa=b, aabbb=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.

1aa2a3a4a5a6a7a8a9a10
11aa2a3a4a5a6a7a8a9a10
aaa2a3a4a5a6a7a8a9a10a3
a2a2a3a4a5a6a7a8a9a10a3a4
a3a3a4a5a6a7a8a9a10a3a4a5
a4a4a5a6a7a8a9a10a3a4a5a6
a5a5a6a7a8a9a10a3a4a5a6a7
a6a6a7a8a9a10a3a4a5a6a7a8
a7a7a8a9a10a3a4a5a6a7a8a9
a8a8a9a10a3a4a5a6a7a8a9a10
a9a9a10a3a4a5a6a7a8a9a10a3
a10a10a3a4a5a6a7a8a9a10a3a4

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. a11 ⇒ a3 [2]
2. b ⇒ a3 [1]
# ab:aaa=b,aabbb=b a/b - -
aaaaaaaaaaa=aaa
b=aaa

Same cardinality

29 unique, 469 total

Σ#PresentationPropertiesDescriptionφ
91472⟨a, b | aaa=bb, abbb=1⟩Grp Com 11ℤ11259
104155⟨a, b | abb=aaa, bba=b⟩Noc 114
105086⟨a, b | aaa=bb, abbb=a⟩Noc Com 1111
105087⟨a, b | aaa=bb, abbb=b⟩Noc Com 113
105111⟨a, b | aab=aa, baaa=b⟩Noc 117
105322⟨a, b | aaa=ab, abb=bb⟩Noc 113
105335⟨a, b | aaa=ab, bbb=aa⟩Noc 113
105410⟨a, b | aab=bb, bab=aa⟩Noc 112
106292⟨a, b | aaa=b, aabbb=a⟩Noc Com 11ℕ(11 = 1)35
106380⟨a, b | aab=a, bbbbb=a⟩Noc Com 11ℕ(11 = 5)1
106834⟨a, b | aba=b, abb=aaa⟩Noc 11
107114⟨a, b | ab=aa, bbbb=aa⟩Noc 111
107116⟨a, b | ab=aa, bbbb=ba⟩Noc 11
108691⟨a, b | aa=b, abbbbb=b⟩Noc Com 11ℕ(11 = 2)32
108777⟨a, b | ab=a, baaaaa=b⟩Noc 1125
109083⟨a, b | ab=a, bbaaa=bb⟩Noc 114
1112179⟨a, b | aaaa=ab, abbb=b⟩Noc Com 11ℕ(11 = 4)20
1115607⟨a, b | aab=aa, bbbbb=a⟩Noc Com 11ℕ(11 = 10)1
1115671⟨a, b | aab=ab, bbbbb=a⟩Noc Com 11ℕ(11 = 6)8
1116148⟨a, b | aab=ab, aaaa=bb⟩Noc 111
1116212⟨a, b | aab=ba, aaaa=bb⟩Noc 11
1118715⟨a, b | aaa=a, aabbba=b⟩Noc 11
1119070⟨a, b | aab=b, bbabbb=a⟩Noc Com 114
1120096⟨a, b | aab=b, abba=aaa⟩Noc 111
1120112⟨a, b | aab=b, baaa=aaa⟩Noc 11
1120168⟨a, b | aab=b, bbbb=aaa⟩Noc Com 11
1120885⟨a, b | bb=aa, aaaaa=ab⟩Noc 1115
1125539⟨a, b | ab=a, baaaa=bbb⟩Noc 11
1125635⟨a, b | ab=a, bbbaa=bbb⟩Noc 11

Isomorphic instances

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

11 total

Σ#PresentationMapping
106299⟨a, b | aaa=b, ababb=b⟩φ(a) = a, φ(b) = aaa
106301⟨a, b | aaa=b, abbab=b⟩φ(a) = a, φ(b) = aaa
106303⟨a, b | aaa=b, abbba=b⟩φ(a) = a, φ(b) = aaa
106309⟨a, b | aaa=b, baabb=b⟩φ(a) = a, φ(b) = aaa
106311⟨a, b | aaa=b, babab=b⟩φ(a) = a, φ(b) = aaa
1124821⟨a, b | aa=b, abbbbb=ab⟩φ(a) = a, φ(b) = aa
1124822⟨a, b | aa=b, abbbbb=ba⟩φ(a) = a, φ(b) = aa
1124847⟨a, b | aa=b, babbbb=ab⟩φ(a) = a, φ(b) = aa
1124848⟨a, b | aa=b, babbbb=ba⟩φ(a) = a, φ(b) = aa
1124854⟨a, b | aa=b, bbabbb=ab⟩φ(a) = a, φ(b) = aa
1124855⟨a, b | aa=b, bbabbb=ba⟩φ(a) = a, φ(b) = aa