#15671 ⟨a, b | aab=ab, bbbbb=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.

1bb2b3b4b5b6b7b8b9b10
11bb2b3b4b5b6b7b8b9b10
bbb2b3b4b5b6b7b8b9b10b6
b2b2b3b4b5b6b7b8b9b10b6b7
b3b3b4b5b6b7b8b9b10b6b7b8
b4b4b5b6b7b8b9b10b6b7b8b9
b5b5b6b7b8b9b10b6b7b8b9b10
b6b6b7b8b9b10b6b7b8b9b10b6
b7b7b8b9b10b6b7b8b9b10b6b7
b8b8b9b10b6b7b8b9b10b6b7b8
b9b9b10b6b7b8b9b10b6b7b8b9
b10b10b6b7b8b9b10b6b7b8b9b10

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. b11 ⇒ b6 [4]
2. a ⇒ b5 [2]
# ab:aab=ab,bbbbb=a b/a - -
bbbbbbbbbbb=bbbbbb
a=bbbbb

Same cardinality

29 unique, 472 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
106293⟨a, b | aaa=b, aabbb=b⟩Noc Com 11ℕ(11 = 3)11
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
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.

8 total

Σ#PresentationMapping
1115735⟨a, b | aab=ba, bbbbb=a⟩φ(a) = bbbbb, φ(b) = b
1115903⟨a, b | aba=ab, bbbbb=a⟩φ(a) = bbbbb, φ(b) = b
1119333⟨a, b | aaa=b, aabbb=bb⟩φ(a) = b, φ(b) = bbb
1119344⟨a, b | aaa=b, ababb=bb⟩φ(a) = b, φ(b) = bbb
1119348⟨a, b | aaa=b, abbab=bb⟩φ(a) = b, φ(b) = bbb
1119351⟨a, b | aaa=b, abbba=bb⟩φ(a) = b, φ(b) = bbb
1119362⟨a, b | aaa=b, baabb=bb⟩φ(a) = b, φ(b) = bbb
1119365⟨a, b | aaa=b, babab=bb⟩φ(a) = b, φ(b) = bbb