#245 ⟨a, b | aa=a, 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.

1bb2b3b4b5
11bb2b3b4b5
bbb2b3b4b5b3
b2b2b3b4b5b3b4
b3b3b4b5b3b4b5
b4b4b5b3b4b5b3
b5b5b3b4b5b3b4

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. b6 ⇒ b3 [4]
2. a ⇒ b3 [2]
# ab:aa=a,bbb=a b/a - -
bbbbbb=bbb
a=bbb

Same cardinality

19 unique, 1876 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
7257⟨a, b | aa=b, bbb=a⟩Noc Com 6ℕ(6 = 1)61
7258⟨a, b | aa=b, bbb=b⟩Noc Com 6ℕ(6 = 2)55
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.

49 total

Σ#PresentationMapping
8969⟨a, b | aa=a, bbb=aa⟩φ(a) = bbb, φ(b) = b
8990⟨a, b | aa=b, bbb=ab⟩φ(a) = b, φ(b) = bb
91591⟨a, b | aaa=ab, abb=b⟩φ(a) = b, φ(b) = bbbbb
92003⟨a, b | aaa=b, aaab=b⟩φ(a) = b, φ(b) = bbb
92005⟨a, b | aaa=b, aaba=b⟩φ(a) = b, φ(b) = bbb
93048⟨a, b | aa=b, aabb=ab⟩φ(a) = b, φ(b) = bb
93049⟨a, b | aa=b, aabb=ba⟩φ(a) = b, φ(b) = bb
93052⟨a, b | aa=b, abab=ab⟩φ(a) = b, φ(b) = bb
93053⟨a, b | aa=b, abab=ba⟩φ(a) = b, φ(b) = bb
93056⟨a, b | aa=b, abba=ab⟩φ(a) = b, φ(b) = bb
93063⟨a, b | aa=b, baab=ab⟩φ(a) = b, φ(b) = bb
93149⟨a, b | aa=a, bbb=aaa⟩φ(a) = bbb, φ(b) = b
93167⟨a, b | aa=b, bbb=aaa⟩φ(a) = b, φ(b) = bb
105045⟨a, b | aaa=ab, aaab=b⟩φ(a) = b, φ(b) = bbbbb
105047⟨a, b | aaa=ab, aaba=b⟩φ(a) = b, φ(b) = bbbbb
105051⟨a, b | aaa=ab, abaa=b⟩φ(a) = b, φ(b) = bbbbb
108916⟨a, b | aa=b, aaaab=ab⟩φ(a) = b, φ(b) = bb
108917⟨a, b | aa=b, aaaab=ba⟩φ(a) = b, φ(b) = bb
108920⟨a, b | aa=b, aaaba=ab⟩φ(a) = b, φ(b) = bb
108921⟨a, b | aa=b, aaaba=ba⟩φ(a) = b, φ(b) = bb
108928⟨a, b | aa=b, aabaa=ab⟩φ(a) = b, φ(b) = bb
109117⟨a, b | aa=a, aaaa=bbb⟩φ(a) = bbb, φ(b) = b
109206⟨a, b | aa=b, aabb=aaa⟩φ(a) = b, φ(b) = bb
109214⟨a, b | aa=b, abab=aaa⟩φ(a) = b, φ(b) = bb
109222⟨a, b | aa=b, abba=aaa⟩φ(a) = b, φ(b) = bb
109236⟨a, b | aa=b, baab=aaa⟩φ(a) = b, φ(b) = bb
1112881⟨a, b | aab=aaa, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1112883⟨a, b | aab=aaa, aaab=b⟩φ(a) = b, φ(b) = bbbb
1112885⟨a, b | aab=aaa, aaba=b⟩φ(a) = b, φ(b) = bbbb
1112887⟨a, b | aab=aaa, aabb=b⟩φ(a) = b, φ(b) = bbbb
1112889⟨a, b | aab=aaa, abaa=b⟩φ(a) = b, φ(b) = bbbb
1112891⟨a, b | aab=aaa, abab=b⟩φ(a) = b, φ(b) = bbbb
1112893⟨a, b | aab=aaa, abba=b⟩φ(a) = b, φ(b) = bbbb
1112895⟨a, b | aab=aaa, abbb=b⟩φ(a) = b, φ(b) = bbbb
1112913⟨a, b | aba=aaa, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1112917⟨a, b | aba=aaa, aaba=b⟩φ(a) = b, φ(b) = bbbb
1112923⟨a, b | aba=aaa, abba=b⟩φ(a) = b, φ(b) = bbbb
1115442⟨a, b | aaa=ab, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1118761⟨a, b | aaa=b, aaaaaa=b⟩φ(a) = b, φ(b) = bbb
1119854⟨a, b | aaa=b, aaab=aaa⟩φ(a) = b, φ(b) = bbb
1119862⟨a, b | aaa=b, aaba=aaa⟩φ(a) = b, φ(b) = bbb
1119951⟨a, b | aab=a, aabb=bbb⟩φ(a) = bbbbb, φ(b) = b
1119967⟨a, b | aab=a, abab=bbb⟩φ(a) = bbbbb, φ(b) = b
1119975⟨a, b | aab=a, abba=bbb⟩φ(a) = bbbbb, φ(b) = b
1124725⟨a, b | aa=b, aaaaaa=ab⟩φ(a) = b, φ(b) = bb
1125121⟨a, b | aa=a, aaaaa=bbb⟩φ(a) = bbb, φ(b) = b
1125266⟨a, b | aa=b, aaaab=aaa⟩φ(a) = b, φ(b) = bb
1125274⟨a, b | aa=b, aaaba=aaa⟩φ(a) = b, φ(b) = bb
1125290⟨a, b | aa=b, aabaa=aaa⟩φ(a) = b, φ(b) = bb