#578 ⟨a, b | aaa=b, abb=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.

1aa2a3a4a5a6
11aa2a3a4a5a6
aaa2a3a4a5a6a3
a2a2a3a4a5a6a3a4
a3a3a4a5a6a3a4a5
a4a4a5a6a3a4a5a6
a5a5a6a3a4a5a6a3
a6a6a3a4a5a6a3a4

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. a7 ⇒ a3 [2]
2. b ⇒ a3 [1]
# ab:aaa=b,abb=b a/b - -
aaaaaaa=aaa
b=aaa

Same cardinality

22 unique, 1179 total

Σ#PresentationPropertiesDescriptionφ
7116⟨a, b | aaa=b, abb=1⟩Grp Com 7ℤ7763
8577⟨a, b | aaa=b, abb=a⟩Noc Com 7ℕ(7 = 1)59
8913⟨a, b | aa=b, abbb=b⟩Noc Com 7ℕ(7 = 2)82
8937⟨a, b | ab=a, baaa=b⟩Noc 766
91593⟨a, b | aaa=ab, baa=b⟩Noc 73
91601⟨a, b | aaa=bb, aab=b⟩Noc Com 75
91620⟨a, b | aab=aa, bbb=a⟩Noc Com 7ℕ(7 = 6)35
91632⟨a, b | aab=ab, bbb=a⟩Noc Com 7ℕ(7 = 4)50
92074⟨a, b | aab=b, babb=a⟩Noc Com 720
92231⟨a, b | ab=aa, aaa=bb⟩Noc 73
92271⟨a, b | bb=aa, aaa=ab⟩Noc 73
93197⟨a, b | ab=a, bbb=baa⟩Noc 75
104162⟨a, b | abb=aab, bbb=a⟩Noc Com 7ℕ(7 = 5)42
106251⟨a, b | aaa=a, aabba=b⟩Noc 71
108987⟨a, b | ab=a, aaaaa=bb⟩Noc Com 75
109108⟨a, b | ab=a, bbbbb=aa⟩Noc Com 72
109110⟨a, b | ab=a, bbbbb=ba⟩Noc 71
109263⟨a, b | ab=a, aaaa=bbb⟩Noc Com 74
109375⟨a, b | ab=a, bbba=bbb⟩Noc 73
109376⟨a, b | ab=a, bbbb=aaa⟩Noc Com 73
109382⟨a, b | ab=a, bbbb=bba⟩Noc 71
1114843⟨a, b | abba=b, aabab=a⟩Noc 71

Isomorphic instances

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

44 total

Σ#PresentationMapping
8580⟨a, b | aaa=b, bab=b⟩φ(a) = a, φ(b) = aaa
93059⟨a, b | aa=b, abbb=ab⟩φ(a) = a, φ(b) = aa
93060⟨a, b | aa=b, abbb=ba⟩φ(a) = a, φ(b) = aa
93066⟨a, b | aa=b, babb=ab⟩φ(a) = a, φ(b) = aa
93067⟨a, b | aa=b, babb=ba⟩φ(a) = a, φ(b) = aa
104185⟨a, b | bab=aaa, bba=b⟩φ(a) = a, φ(b) = aaa
105049⟨a, b | aaa=ab, aabb=b⟩φ(a) = a, φ(b) = aaaaaa
105053⟨a, b | aaa=ab, abab=b⟩φ(a) = a, φ(b) = aaaaaa
105055⟨a, b | aaa=ab, abba=b⟩φ(a) = a, φ(b) = aaaaaa
106281⟨a, b | aaa=b, aaaab=b⟩φ(a) = a, φ(b) = aaa
106283⟨a, b | aaa=b, aaaba=b⟩φ(a) = a, φ(b) = aaa
106287⟨a, b | aaa=b, aabaa=b⟩φ(a) = a, φ(b) = aaa
106805⟨a, b | aaa=b, abb=aaa⟩φ(a) = a, φ(b) = aaa
106806⟨a, b | aaa=b, bab=aaa⟩φ(a) = a, φ(b) = aaa
106816⟨a, b | aab=a, bbb=aab⟩φ(a) = aaa, φ(b) = a
106833⟨a, b | aba=a, bbb=aba⟩φ(a) = aaa, φ(b) = a
108924⟨a, b | aa=b, aaabb=ab⟩φ(a) = a, φ(b) = aa
108925⟨a, b | aa=b, aaabb=ba⟩φ(a) = a, φ(b) = aa
108931⟨a, b | aa=b, aabab=ab⟩φ(a) = a, φ(b) = aa
108932⟨a, b | aa=b, aabab=ba⟩φ(a) = a, φ(b) = aa
108935⟨a, b | aa=b, aabba=ab⟩φ(a) = a, φ(b) = aa
108936⟨a, b | aa=b, aabba=ba⟩φ(a) = a, φ(b) = aa
108943⟨a, b | aa=b, abaab=ab⟩φ(a) = a, φ(b) = aa
108944⟨a, b | aa=b, abaab=ba⟩φ(a) = a, φ(b) = aa
108947⟨a, b | aa=b, ababa=ab⟩φ(a) = a, φ(b) = aa
108965⟨a, b | aa=b, baaab=ab⟩φ(a) = a, φ(b) = aa
109228⟨a, b | aa=b, abbb=aaa⟩φ(a) = a, φ(b) = aa
109242⟨a, b | aa=b, babb=aaa⟩φ(a) = a, φ(b) = aa
1115444⟨a, b | aaa=ab, aaaab=b⟩φ(a) = a, φ(b) = aaaaaa
1115446⟨a, b | aaa=ab, aaaba=b⟩φ(a) = a, φ(b) = aaaaaa
1115450⟨a, b | aaa=ab, aabaa=b⟩φ(a) = a, φ(b) = aaaaaa
1115458⟨a, b | aaa=ab, abaaa=b⟩φ(a) = a, φ(b) = aaaaaa
1124728⟨a, b | aa=b, aaaaab=ab⟩φ(a) = a, φ(b) = aa
1124729⟨a, b | aa=b, aaaaab=ba⟩φ(a) = a, φ(b) = aa
1124732⟨a, b | aa=b, aaaaba=ab⟩φ(a) = a, φ(b) = aa
1124733⟨a, b | aa=b, aaaaba=ba⟩φ(a) = a, φ(b) = aa
1124740⟨a, b | aa=b, aaabaa=ab⟩φ(a) = a, φ(b) = aa
1124741⟨a, b | aa=b, aaabaa=ba⟩φ(a) = a, φ(b) = aa
1125282⟨a, b | aa=b, aaabb=aaa⟩φ(a) = a, φ(b) = aa
1125296⟨a, b | aa=b, aabab=aaa⟩φ(a) = a, φ(b) = aa
1125304⟨a, b | aa=b, aabba=aaa⟩φ(a) = a, φ(b) = aa
1125320⟨a, b | aa=b, abaab=aaa⟩φ(a) = a, φ(b) = aa
1125328⟨a, b | aa=b, ababa=aaa⟩φ(a) = a, φ(b) = aa
1125364⟨a, b | aa=b, baaab=aaa⟩φ(a) = a, φ(b) = aa