#1605 ⟨a, b | aaa=bb, 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.

1aa2a3a4a5a6a7
11aa2a3a4a5a6a7
aaa2a3a4a5a6a7a3
a2a2a3a4a5a6a7a3a4
a3a3a4a5a6a7a3a4a5
a4a4a5a6a7a3a4a5a6
a5a5a6a7a3a4a5a6a7
a6a6a7a3a4a5a6a7a3
a7a7a3a4a5a6a7a3a4

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. a8 ⇒ a3 [3]
2. b ⇒ a4 [2]
# ab:aaa=bb,abb=b a/b - -
aaaaaaaa=aaa
b=aaaa

Same cardinality

33 unique, 1156 total

Σ#PresentationPropertiesDescriptionφ
7188⟨a, b | aab=1, bbbb=1⟩Grp Com 8ℤ8727
7330⟨a, b | aa=1, abba=b⟩Noc 860
8556⟨a, b | aba=b, aabb=1⟩Grp 828
8898⟨a, b | aa=a, bbbb=a⟩Noc Com 8ℕ(8 = 4)48
8918⟨a, b | aa=b, bbbb=a⟩Noc Com 8ℕ(8 = 1)34
8919⟨a, b | aa=b, bbbb=b⟩Noc Com 8ℕ(8 = 2)46
8961⟨a, b | aa=a, aba=bb⟩Noc 85
91606⟨a, b | aaa=bb, bab=a⟩Noc Com 814
91615⟨a, b | aab=aa, baa=b⟩Noc 814
91650⟨a, b | aab=bb, baa=a⟩Noc 815
92206⟨a, b | ab=aa, bbbb=a⟩Noc Com 8ℕ(8 = 5)32
92220⟨a, b | bb=aa, aaab=a⟩Noc Com 822
92247⟨a, b | ab=aa, baa=bb⟩Noc 81
92256⟨a, b | ab=aa, bbb=aa⟩Noc 81
92258⟨a, b | ab=aa, bbb=ba⟩Noc 8
92883⟨a, b | aa=a, abbbb=b⟩Noc 814
93107⟨a, b | ab=a, baaa=bb⟩Noc 89
93123⟨a, b | ab=a, bbaa=bb⟩Noc 810
105191⟨a, b | aab=bb, aaaa=b⟩Noc Com 8ℕ(8 = 6)18
106664⟨a, b | aab=b, aaaa=ba⟩Noc 81
107057⟨a, b | ab=aa, aaaa=bb⟩Noc 87
109380⟨a, b | ab=a, bbbb=baa⟩Noc 82
1112606⟨a, b | abbb=aa, bbbb=a⟩Noc Com 8ℕ(8 = 7)14
1120047⟨a, b | aab=a, bbbb=bbb⟩Noc 8
1120051⟨a, b | aab=b, aaaa=abb⟩Noc Com 81
1124863⟨a, b | ab=a, aaaaaa=bb⟩Noc Com 8
1125112⟨a, b | ab=a, bbbbbb=aa⟩Noc Com 8
1125114⟨a, b | ab=a, bbbbbb=ba⟩Noc 8
1125411⟨a, b | ab=a, aaaaa=bbb⟩Noc Com 8
1125652⟨a, b | ab=a, bbbbb=aaa⟩Noc Com 8
1125658⟨a, b | ab=a, bbbbb=bba⟩Noc 8
1125897⟨a, b | ab=a, bbbb=aaaa⟩Noc Com 8
1125911⟨a, b | ab=a, bbbb=bbba⟩Noc 8

Isomorphic instances

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

55 total

Σ#PresentationMapping
92007⟨a, b | aaa=b, aabb=b⟩φ(a) = a, φ(b) = aaa
92009⟨a, b | aaa=b, abab=b⟩φ(a) = a, φ(b) = aaa
92011⟨a, b | aaa=b, abba=b⟩φ(a) = a, φ(b) = aaa
92015⟨a, b | aaa=b, baab=b⟩φ(a) = a, φ(b) = aaa
93070⟨a, b | aa=b, bbbb=ab⟩φ(a) = a, φ(b) = aa
105057⟨a, b | aaa=ab, abbb=b⟩φ(a) = a, φ(b) = aaaaaaa
105075⟨a, b | aaa=bb, aaaa=b⟩φ(a) = a, φ(b) = aaaa
108939⟨a, b | aa=b, aabbb=ab⟩φ(a) = a, φ(b) = aa
108940⟨a, b | aa=b, aabbb=ba⟩φ(a) = a, φ(b) = aa
108950⟨a, b | aa=b, ababb=ab⟩φ(a) = a, φ(b) = aa
108951⟨a, b | aa=b, ababb=ba⟩φ(a) = a, φ(b) = aa
108954⟨a, b | aa=b, abbab=ab⟩φ(a) = a, φ(b) = aa
108955⟨a, b | aa=b, abbab=ba⟩φ(a) = a, φ(b) = aa
108958⟨a, b | aa=b, abbba=ab⟩φ(a) = a, φ(b) = aa
108968⟨a, b | aa=b, baabb=ab⟩φ(a) = a, φ(b) = aa
108969⟨a, b | aa=b, baabb=ba⟩φ(a) = a, φ(b) = aa
108972⟨a, b | aa=b, babab=ab⟩φ(a) = a, φ(b) = aa
109250⟨a, b | aa=b, bbbb=aaa⟩φ(a) = a, φ(b) = aa
1115448⟨a, b | aaa=ab, aaabb=b⟩φ(a) = a, φ(b) = aaaaaaa
1115452⟨a, b | aaa=ab, aabab=b⟩φ(a) = a, φ(b) = aaaaaaa
1115454⟨a, b | aaa=ab, aabba=b⟩φ(a) = a, φ(b) = aaaaaaa
1115460⟨a, b | aaa=ab, abaab=b⟩φ(a) = a, φ(b) = aaaaaaa
1115462⟨a, b | aaa=ab, ababa=b⟩φ(a) = a, φ(b) = aaaaaaa
1115466⟨a, b | aaa=ab, abbaa=b⟩φ(a) = a, φ(b) = aaaaaaa
1118763⟨a, b | aaa=b, aaaaab=b⟩φ(a) = a, φ(b) = aaa
1118765⟨a, b | aaa=b, aaaaba=b⟩φ(a) = a, φ(b) = aaa
1118769⟨a, b | aaa=b, aaabaa=b⟩φ(a) = a, φ(b) = aaa
1119870⟨a, b | aaa=b, aabb=aaa⟩φ(a) = a, φ(b) = aaa
1119878⟨a, b | aaa=b, abab=aaa⟩φ(a) = a, φ(b) = aaa
1119886⟨a, b | aaa=b, abba=aaa⟩φ(a) = a, φ(b) = aaa
1119900⟨a, b | aaa=b, baab=aaa⟩φ(a) = a, φ(b) = aaa
1119935⟨a, b | aab=a, aaab=bbb⟩φ(a) = aaaa, φ(b) = a
1119943⟨a, b | aab=a, aaba=bbb⟩φ(a) = aaaa, φ(b) = a
1119959⟨a, b | aab=a, abaa=bbb⟩φ(a) = aaaa, φ(b) = a
1124736⟨a, b | aa=b, aaaabb=ab⟩φ(a) = a, φ(b) = aa
1124737⟨a, b | aa=b, aaaabb=ba⟩φ(a) = a, φ(b) = aa
1124744⟨a, b | aa=b, aaabab=ab⟩φ(a) = a, φ(b) = aa
1124745⟨a, b | aa=b, aaabab=ba⟩φ(a) = a, φ(b) = aa
1124748⟨a, b | aa=b, aaabba=ab⟩φ(a) = a, φ(b) = aa
1124749⟨a, b | aa=b, aaabba=ba⟩φ(a) = a, φ(b) = aa
1124756⟨a, b | aa=b, aabaab=ab⟩φ(a) = a, φ(b) = aa
1124757⟨a, b | aa=b, aabaab=ba⟩φ(a) = a, φ(b) = aa
1124760⟨a, b | aa=b, aababa=ab⟩φ(a) = a, φ(b) = aa
1124761⟨a, b | aa=b, aababa=ba⟩φ(a) = a, φ(b) = aa
1124768⟨a, b | aa=b, aabbaa=ab⟩φ(a) = a, φ(b) = aa
1124783⟨a, b | aa=b, abaaab=ab⟩φ(a) = a, φ(b) = aa
1124784⟨a, b | aa=b, abaaab=ba⟩φ(a) = a, φ(b) = aa
1124787⟨a, b | aa=b, abaaba=ab⟩φ(a) = a, φ(b) = aa
1124825⟨a, b | aa=b, baaaab=ab⟩φ(a) = a, φ(b) = aa
1125312⟨a, b | aa=b, aabbb=aaa⟩φ(a) = a, φ(b) = aa
1125334⟨a, b | aa=b, ababb=aaa⟩φ(a) = a, φ(b) = aa
1125342⟨a, b | aa=b, abbab=aaa⟩φ(a) = a, φ(b) = aa
1125350⟨a, b | aa=b, abbba=aaa⟩φ(a) = a, φ(b) = aa
1125370⟨a, b | aa=b, baabb=aaa⟩φ(a) = a, φ(b) = aa
1125378⟨a, b | aa=b, babab=aaa⟩φ(a) = a, φ(b) = aa