#258 ⟨a, b | aa=b, bbb=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.

1aa2a3a4a5
11aa2a3a4a5
aaa2a3a4a5a2
a2a2a3a4a5a2a3
a3a3a4a5a2a3a4
a4a4a5a2a3a4a5
a5a5a2a3a4a5a2

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. a6 ⇒ a2 [2]
2. b ⇒ a2 [1]
# ab:aa=b,bbb=b a/b - -
aaaaaa=aa
b=aa

Same cardinality

19 unique, 1870 total

Σ#PresentationPropertiesDescriptionφ
622⟨a, b | ab=aa, bb=1⟩Noc 6107
626⟨a, b | aaa=1, abb=1⟩Grp Com 6ℤ61373
7160⟨a, b | ab=aa, bb=b⟩Noc 64
7245⟨a, b | aa=a, bbb=a⟩Noc Com 6ℕ(6 = 3)49
7257⟨a, b | aa=b, bbb=a⟩Noc Com 6ℕ(6 = 1)61
8639⟨a, b | ab=aa, baa=b⟩Noc 613
8644⟨a, b | ab=aa, bbb=a⟩Noc Com 6ℕ(6 = 4)46
8893⟨a, b | aa=a, abbb=b⟩Noc 619
81011⟨a, b | ab=a, baa=bb⟩Noc 613
91427⟨a, b | abba=b, baba=1⟩Grp 666
91581⟨a, b | aaa=aa, abb=b⟩Noc 65
91648⟨a, b | aab=bb, abb=a⟩Noc Com 613
91686⟨a, b | abb=aa, bbb=a⟩Noc Com 6ℕ(6 = 5)49
93075⟨a, b | ab=a, aaaa=bb⟩Noc Com 614
93132⟨a, b | ab=a, bbbb=aa⟩Noc Com 65
93134⟨a, b | ab=a, bbbb=ba⟩Noc 62
93193⟨a, b | ab=a, bbb=aaa⟩Noc Com 69
93199⟨a, b | ab=a, bbb=bba⟩Noc 63
1124145⟨a, b | aa=a, abbbbba=b⟩Noc Com 6

Isomorphic instances

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

55 total

Σ#PresentationMapping
8653⟨a, b | bb=aa, aaa=b⟩φ(a) = a, φ(b) = aaa
8654⟨a, b | bb=aa, aab=a⟩φ(a) = aaa, φ(b) = a
8907⟨a, b | aa=b, aabb=b⟩φ(a) = a, φ(b) = aa
8909⟨a, b | aa=b, abab=b⟩φ(a) = a, φ(b) = aa
8911⟨a, b | aa=b, abba=b⟩φ(a) = a, φ(b) = aa
8915⟨a, b | aa=b, baab=b⟩φ(a) = a, φ(b) = aa
8989⟨a, b | aa=b, bbb=aa⟩φ(a) = a, φ(b) = aa
92899⟨a, b | aa=b, aaaab=b⟩φ(a) = a, φ(b) = aa
92901⟨a, b | aa=b, aaaba=b⟩φ(a) = a, φ(b) = aa
92905⟨a, b | aa=b, aabaa=b⟩φ(a) = a, φ(b) = aa
93047⟨a, b | aa=b, aabb=aa⟩φ(a) = a, φ(b) = aa
93051⟨a, b | aa=b, abab=aa⟩φ(a) = a, φ(b) = aa
93055⟨a, b | aa=b, abba=aa⟩φ(a) = a, φ(b) = aa
93062⟨a, b | aa=b, baab=aa⟩φ(a) = a, φ(b) = aa
105095⟨a, b | aab=aa, aaaa=b⟩φ(a) = a, φ(b) = aaaa
105223⟨a, b | aba=aa, aaaa=b⟩φ(a) = a, φ(b) = aaaa
106565⟨a, b | aaa=b, aaab=aa⟩φ(a) = a, φ(b) = aaa
106569⟨a, b | aaa=b, aaba=aa⟩φ(a) = a, φ(b) = aaa
106605⟨a, b | aab=a, aaab=bb⟩φ(a) = aaa, φ(b) = a
106609⟨a, b | aab=a, aaba=bb⟩φ(a) = aaa, φ(b) = a
106617⟨a, b | aab=a, abaa=bb⟩φ(a) = aaa, φ(b) = a
106951⟨a, b | ab=aa, aaaaa=b⟩φ(a) = a, φ(b) = aaaaa
106953⟨a, b | ab=aa, aaaab=b⟩φ(a) = a, φ(b) = aaaaa
106955⟨a, b | ab=aa, aaaba=b⟩φ(a) = a, φ(b) = aaaaa
106957⟨a, b | ab=aa, aaabb=b⟩φ(a) = a, φ(b) = aaaaa
106959⟨a, b | ab=aa, aabaa=b⟩φ(a) = a, φ(b) = aaaaa
106961⟨a, b | ab=aa, aabab=b⟩φ(a) = a, φ(b) = aaaaa
106963⟨a, b | ab=aa, aabba=b⟩φ(a) = a, φ(b) = aaaaa
106965⟨a, b | ab=aa, aabbb=b⟩φ(a) = a, φ(b) = aaaaa
106967⟨a, b | ab=aa, abaaa=b⟩φ(a) = a, φ(b) = aaaaa
106969⟨a, b | ab=aa, abaab=b⟩φ(a) = a, φ(b) = aaaaa
106971⟨a, b | ab=aa, ababa=b⟩φ(a) = a, φ(b) = aaaaa
106973⟨a, b | ab=aa, ababb=b⟩φ(a) = a, φ(b) = aaaaa
106975⟨a, b | ab=aa, abbaa=b⟩φ(a) = a, φ(b) = aaaaa
106977⟨a, b | ab=aa, abbab=b⟩φ(a) = a, φ(b) = aaaaa
106979⟨a, b | ab=aa, abbba=b⟩φ(a) = a, φ(b) = aaaaa
106981⟨a, b | ab=aa, abbbb=b⟩φ(a) = a, φ(b) = aaaaa
108641⟨a, b | aa=b, aaaaaa=b⟩φ(a) = a, φ(b) = aa
108915⟨a, b | aa=b, aaaab=aa⟩φ(a) = a, φ(b) = aa
108919⟨a, b | aa=b, aaaba=aa⟩φ(a) = a, φ(b) = aa
108927⟨a, b | aa=b, aabaa=aa⟩φ(a) = a, φ(b) = aa
1112433⟨a, b | aabb=aa, abab=b⟩φ(a) = a, φ(b) = aa
1112435⟨a, b | aabb=aa, abba=b⟩φ(a) = a, φ(b) = aa
1112596⟨a, b | abba=bb, baab=a⟩φ(a) = aa, φ(b) = a
1112598⟨a, b | abba=bb, baba=a⟩φ(a) = aa, φ(b) = a
1112602⟨a, b | abba=bb, bbaa=a⟩φ(a) = aa, φ(b) = a
1112629⟨a, b | baba=aa, bbaa=b⟩φ(a) = a, φ(b) = aa
1115548⟨a, b | aab=aa, aaaab=b⟩φ(a) = a, φ(b) = aaaa
1115550⟨a, b | aab=aa, aaaba=b⟩φ(a) = a, φ(b) = aaaa
1115554⟨a, b | aab=aa, aabaa=b⟩φ(a) = a, φ(b) = aaaa
1115562⟨a, b | aab=aa, abaaa=b⟩φ(a) = a, φ(b) = aaaa
1115806⟨a, b | aba=aa, aaaba=b⟩φ(a) = a, φ(b) = aaaa
1115810⟨a, b | aba=aa, aabaa=b⟩φ(a) = a, φ(b) = aaaa
1119234⟨a, b | aaa=a, aaaaa=bb⟩φ(a) = aa, φ(b) = a
1124724⟨a, b | aa=b, aaaaaa=aa⟩φ(a) = a, φ(b) = aa