#581 ⟨a, b | aaa=b, bbb=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.

1aa2a3a4a5a6a7a8
11aa2a3a4a5a6a7a8
aaa2a3a4a5a6a7a8a
a2a2a3a4a5a6a7a8aa2
a3a3a4a5a6a7a8aa2a3
a4a4a5a6a7a8aa2a3a4
a5a5a6a7a8aa2a3a4a5
a6a6a7a8aa2a3a4a5a6
a7a7a8aa2a3a4a5a6a7
a8a8aa2a3a4a5a6a7a8

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. a9 ⇒ a [2]
2. b ⇒ a3 [1]
# ab:aaa=b,bbb=a a/b - -
aaaaaaaaa=a
b=aaa

Same cardinality

26 unique, 880 total

Σ#PresentationPropertiesDescriptionφ
7118⟨a, b | aaa=b, bbb=1⟩Grp Com 9ℤ9562
8568⟨a, b | aaa=a, abb=b⟩Noc 95
8571⟨a, b | aaa=a, bbb=a⟩Noc Com 9ℕ(9 = 3)48
8601⟨a, b | aba=b, bab=a⟩Noc 93
91584⟨a, b | aaa=aa, bbb=a⟩Noc Com 9ℕ(9 = 6)16
91595⟨a, b | aaa=ab, bab=b⟩Noc 99
91598⟨a, b | aaa=ab, bbb=a⟩Noc Com 9ℕ(9 = 4)40
91608⟨a, b | aaa=bb, bbb=a⟩Noc Com 9ℕ(9 = 2)70
91991⟨a, b | aaa=a, abba=b⟩Noc 93
92064⟨a, b | aab=b, abba=a⟩Noc 938
92255⟨a, b | ab=aa, bba=bb⟩Noc 9
104125⟨a, b | aab=aaa, baa=b⟩Noc 92
104130⟨a, b | aab=aaa, bbb=a⟩Noc Com 9ℕ(9 = 7)1
104156⟨a, b | abb=aaa, bbb=a⟩Noc Com 9ℕ(9 = 5)12
105035⟨a, b | aaa=aa, abbb=b⟩Noc 93
105079⟨a, b | aaa=bb, aaba=b⟩Noc Com 98
105124⟨a, b | aab=aa, bbbb=a⟩Noc Com 9ℕ(9 = 8)5
105339⟨a, b | aaa=bb, aab=ab⟩Noc 91
106824⟨a, b | aab=b, bbb=aaa⟩Noc Com 95
107139⟨a, b | bb=aa, aaaa=ab⟩Noc 93
109327⟨a, b | ab=a, baaa=bbb⟩Noc 93
109359⟨a, b | ab=a, bbaa=bbb⟩Noc 92
1119506⟨a, b | aab=b, aaaaa=ba⟩Noc 9
1119507⟨a, b | aab=b, aaaaa=bb⟩Noc Com 9
1120723⟨a, b | ab=aa, aaaaa=bb⟩Noc 915
1125656⟨a, b | ab=a, bbbbb=baa⟩Noc 9

Isomorphic instances

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

62 total

Σ#PresentationMapping
92082⟨a, b | aab=b, bbbb=a⟩φ(a) = bbbb, φ(b) = b
92122⟨a, b | aba=b, bbbb=a⟩φ(a) = bbbb, φ(b) = b
92922⟨a, b | aa=b, abbbb=a⟩φ(a) = b, φ(b) = bb
92930⟨a, b | aa=b, babbb=a⟩φ(a) = b, φ(b) = bb
92932⟨a, b | aa=b, bbabb=a⟩φ(a) = b, φ(b) = bb
106284⟨a, b | aaa=b, aaabb=a⟩φ(a) = bbb, φ(b) = b
106288⟨a, b | aaa=b, aabab=a⟩φ(a) = bbb, φ(b) = b
106290⟨a, b | aaa=b, aabba=a⟩φ(a) = bbb, φ(b) = b
106294⟨a, b | aaa=b, abaab=a⟩φ(a) = bbb, φ(b) = b
106296⟨a, b | aaa=b, ababa=a⟩φ(a) = bbb, φ(b) = b
106306⟨a, b | aaa=b, baaab=a⟩φ(a) = bbb, φ(b) = b
108654⟨a, b | aa=b, aaabbb=a⟩φ(a) = b, φ(b) = bb
108660⟨a, b | aa=b, aababb=a⟩φ(a) = b, φ(b) = bb
108664⟨a, b | aa=b, aabbab=a⟩φ(a) = b, φ(b) = bb
108666⟨a, b | aa=b, aabbba=a⟩φ(a) = b, φ(b) = bb
108674⟨a, b | aa=b, abaabb=a⟩φ(a) = b, φ(b) = bb
108676⟨a, b | aa=b, ababab=a⟩φ(a) = b, φ(b) = bb
108678⟨a, b | aa=b, ababba=a⟩φ(a) = b, φ(b) = bb
108682⟨a, b | aa=b, abbaab=a⟩φ(a) = b, φ(b) = bb
108694⟨a, b | aa=b, baaabb=a⟩φ(a) = b, φ(b) = bb
108696⟨a, b | aa=b, baabab=a⟩φ(a) = b, φ(b) = bb
1114668⟨a, b | aabb=a, aaaab=b⟩φ(a) = bbbbbb, φ(b) = b
1114670⟨a, b | aabb=a, aaaba=b⟩φ(a) = bbbbbb, φ(b) = b
1114674⟨a, b | aabb=a, aabaa=b⟩φ(a) = bbbbbb, φ(b) = b
1114682⟨a, b | aabb=a, abaaa=b⟩φ(a) = bbbbbb, φ(b) = b
1114732⟨a, b | abab=a, aaaab=b⟩φ(a) = bbbbbb, φ(b) = b
1114734⟨a, b | abab=a, aaaba=b⟩φ(a) = bbbbbb, φ(b) = b
1114738⟨a, b | abab=a, aabaa=b⟩φ(a) = bbbbbb, φ(b) = b
1114746⟨a, b | abab=a, abaaa=b⟩φ(a) = bbbbbb, φ(b) = b
1114798⟨a, b | abba=a, aaaba=b⟩φ(a) = bbbbbb, φ(b) = b
1114802⟨a, b | abba=a, aabaa=b⟩φ(a) = bbbbbb, φ(b) = b
1114859⟨a, b | abba=b, abbbb=a⟩φ(a) = b, φ(b) = bbbbbb
1114867⟨a, b | abba=b, babbb=a⟩φ(a) = b, φ(b) = bbbbbb
1118990⟨a, b | aab=b, aabbbb=a⟩φ(a) = bbbb, φ(b) = b
1119006⟨a, b | aab=b, ababbb=a⟩φ(a) = bbbb, φ(b) = b
1119014⟨a, b | aab=b, abbabb=a⟩φ(a) = bbbb, φ(b) = b
1119018⟨a, b | aab=b, abbbab=a⟩φ(a) = bbbb, φ(b) = b
1119038⟨a, b | aab=b, baabbb=a⟩φ(a) = bbbb, φ(b) = b
1119046⟨a, b | aab=b, bababb=a⟩φ(a) = bbbb, φ(b) = b
1119050⟨a, b | aab=b, babbab=a⟩φ(a) = bbbb, φ(b) = b
1119062⟨a, b | aab=b, bbaabb=a⟩φ(a) = bbbb, φ(b) = b
1119066⟨a, b | aab=b, bbabab=a⟩φ(a) = bbbb, φ(b) = b
1119074⟨a, b | aab=b, bbbaab=a⟩φ(a) = bbbb, φ(b) = b
1119188⟨a, b | aba=b, aabbbb=a⟩φ(a) = bbbb, φ(b) = b
1119200⟨a, b | aba=b, ababbb=a⟩φ(a) = bbbb, φ(b) = b
1119204⟨a, b | aba=b, abbabb=a⟩φ(a) = bbbb, φ(b) = b
1119206⟨a, b | aba=b, abbbab=a⟩φ(a) = bbbb, φ(b) = b
1119208⟨a, b | aba=b, abbbba=a⟩φ(a) = bbbb, φ(b) = b
1119220⟨a, b | aba=b, bababb=a⟩φ(a) = bbbb, φ(b) = b
1119226⟨a, b | aba=b, bbaabb=a⟩φ(a) = bbbb, φ(b) = b
1124194⟨a, b | aa=b, aaaaabb=a⟩φ(a) = b, φ(b) = bb
1124198⟨a, b | aa=b, aaaabab=a⟩φ(a) = b, φ(b) = bb
1124200⟨a, b | aa=b, aaaabba=a⟩φ(a) = b, φ(b) = bb
1124206⟨a, b | aa=b, aaabaab=a⟩φ(a) = b, φ(b) = bb
1124208⟨a, b | aa=b, aaababa=a⟩φ(a) = b, φ(b) = bb
1124212⟨a, b | aa=b, aaabbaa=a⟩φ(a) = b, φ(b) = bb
1124220⟨a, b | aa=b, aabaaab=a⟩φ(a) = b, φ(b) = bb
1124222⟨a, b | aa=b, aabaaba=a⟩φ(a) = b, φ(b) = bb
1124226⟨a, b | aa=b, aababaa=a⟩φ(a) = b, φ(b) = bb
1124248⟨a, b | aa=b, abaaaab=a⟩φ(a) = b, φ(b) = bb
1124250⟨a, b | aa=b, abaaaba=a⟩φ(a) = b, φ(b) = bb
1124292⟨a, b | aa=b, baaaaab=a⟩φ(a) = b, φ(b) = bb