#14647 ⟨a, b | aaba=b, babbb=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.

1aba2abb2a3a2bab2b3a2b2ab3a2b3
11aba2abb2a3a2bab2b3a2b2ab3a2b3
aaa2aba3a2bab2aba2b2ab3b2a2b3b3
bbabb2a2bab2b3ba2b2ab3a3a2b3aa2
a2a2a3a2baba2b2a2abb2a2b3ab2b3ab3
ababa2bab2ba2b2ab3abb2a2b3ab3a2a3
b2b2ab2b3a2b2ab3a3b2a2b3aba2aba2b
a3a3aba2abb2a3a2bab2b3a2b2ab3a2b3
a2ba2bba2b2abb2a2b3a2bab2b3a2ab3a3a
ab2ab2a2b2ab3b2a2b3aab2b3a2aba3a2bb
b3b3ab3a3a2b3abb3a2abb2a2bab2a2b2
a2b2a2b2b2a2b3ab2b3a2a2b2ab3a3a2babab
ab3ab3a2b3ab3a2abab3a3a2bab2ba2b2b2
a2b3a2b3b3a2ab3a3a2ba2b3aba2b2abb2ab2

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 [12]
2. a4 ⇒ a [13]
3. a3b ⇒ b [11]
4. b4 ⇒ a3 [4]
# ab:aaba=b,babbb=a ab - -
ba=ab
aaaa=a
aaab=b
bbbb=aaa

Same cardinality

26 unique, 301 total

Σ#PresentationPropertiesDescriptionφ
91328⟨a, b | aaaa=b, abbb=1⟩Grp Com 13ℤ13189
92118⟨a, b | aba=b, baab=a⟩Noc 134
104642⟨a, b | aaaa=b, abbb=a⟩Noc Com 13ℕ(13 = 1)10
104643⟨a, b | aaaa=b, abbb=b⟩Noc Com 13ℕ(13 = 4)6
104683⟨a, b | aaab=b, abba=a⟩Noc 1316
105065⟨a, b | aaa=ab, babb=b⟩Noc 137
105334⟨a, b | aaa=ab, bba=bb⟩Noc 13
105336⟨a, b | aaa=ab, bbb=ab⟩Noc 131
105337⟨a, b | aaa=ab, bbb=ba⟩Noc 13
105340⟨a, b | aaa=bb, aab=ba⟩Noc 13
106305⟨a, b | aaa=b, abbbb=b⟩Noc Com 13ℕ(13 = 3)2
107143⟨a, b | bb=aa, aaab=ba⟩Noc 131
1112240⟨a, b | aaab=aa, bbbb=a⟩Noc Com 13ℕ(13 = 8)1
1112268⟨a, b | aaab=ab, bbbb=a⟩Noc Com 13ℕ(13 = 5)3
1112324⟨a, b | aaab=bb, bbbb=a⟩Noc Com 13ℕ(13 = 2)14
1115520⟨a, b | aaa=bb, aabbb=b⟩Noc Com 132
1116012⟨a, b | aaa=ab, abbb=bb⟩Noc 13
1116069⟨a, b | aaa=bb, abbb=ba⟩Noc 131
1116205⟨a, b | aab=ab, bbbb=aa⟩Noc 132
1116459⟨a, b | aba=bb, abbb=aa⟩Noc 131
1116506⟨a, b | aab=ab, bbb=aaa⟩Noc 131
1116515⟨a, b | aab=ba, bbb=aaa⟩Noc 13
1118958⟨a, b | aab=a, bbbbbb=a⟩Noc Com 13ℕ(13 = 6)4
1120046⟨a, b | aab=a, bbbb=bba⟩Noc 13
1120927⟨a, b | ab=aa, aaaa=bbb⟩Noc 137
1120991⟨a, b | ab=aa, baaa=bbb⟩Noc 133

Isomorphic instances

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

2 total

Σ#PresentationMapping
1114655⟨a, b | aaba=b, bbabb=a⟩φ(a) = a, φ(b) = b
1114659⟨a, b | aaba=b, bbbab=a⟩φ(a) = a, φ(b) = b