#5086 ⟨a, b | aaa=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.

1aba2abb2a2bab2b3a2b2b4
11aba2abb2a2bab2b3a2b2b4
aaa2abb2a2bab2b3a2b2ab4ab
bbabb2a2bab2b3a2b2ab4a2b2
a2a2b2a2bab2b3a2b2ab4a2aba2b
ababa2bab2b3a2b2ab4a2abb2ab2
b2b2ab2b3a2b2ab4a2abb2a2bb3
a2ba2bb3a2b2ab4a2abb2a2bab2a2b2
ab2ab2a2b2ab4a2abb2a2bab2b3a
b3b3ab4a2abb2a2bab2b3a2b2b4
a2b2a2b2b4a2abb2a2bab2b3a2b2aa2
b4b4abb2a2bab2b3a2b2ab4a2b2

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. a3 ⇒ b2 [1]
3. ab3 ⇒ a [2]
4. b5 ⇒ b2 [4]
# ab:aaa=bb,abbb=a ab - -
ba=ab
aaa=bb
abbb=a
bbbbb=bb

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
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
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
105090⟨a, b | aaa=bb, babb=a⟩φ(a) = a, φ(b) = b
1112308⟨a, b | aaab=bb, abbb=a⟩φ(a) = aa, φ(b) = b
1112316⟨a, b | aaab=bb, babb=a⟩φ(a) = aa, φ(b) = b
1112320⟨a, b | aaab=bb, bbab=a⟩φ(a) = aa, φ(b) = b
1112412⟨a, b | aaba=bb, abbb=a⟩φ(a) = aa, φ(b) = b
1112420⟨a, b | aaba=bb, babb=a⟩φ(a) = aa, φ(b) = b
1112424⟨a, b | aaba=bb, bbab=a⟩φ(a) = aa, φ(b) = b
1112426⟨a, b | aaba=bb, bbba=a⟩φ(a) = aa, φ(b) = b
1115507⟨a, b | aaa=bb, aaaab=a⟩φ(a) = a, φ(b) = b
1115509⟨a, b | aaa=bb, aaaba=a⟩φ(a) = a, φ(b) = b
1115513⟨a, b | aaa=bb, aabaa=a⟩φ(a) = a, φ(b) = b