#644 ⟨a, b | ab=aa, 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
bbb2b3b4b5b4
b2b2b3b4b5b4b5
b3b3b4b5b4b5b4
b4b4b5b4b5b4b5
b5b5b4b5b4b5b4

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 ⇒ b4 [4]
2. a ⇒ b3 [2]
# ab:ab=aa,bbb=a b/a - -
bbbbbb=bbbb
a=bbb

Same cardinality

19 unique, 1879 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
7258⟨a, b | aa=b, bbb=b⟩Noc Com 6ℕ(6 = 2)55
8639⟨a, b | ab=aa, baa=b⟩Noc 613
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.

46 total

Σ#PresentationMapping
8991⟨a, b | aa=b, bbb=bb⟩φ(a) = b, φ(b) = bb
92053⟨a, b | aab=b, aaaa=b⟩φ(a) = b, φ(b) = bbbb
92105⟨a, b | aba=b, aaaa=b⟩φ(a) = b, φ(b) = bbbb
93050⟨a, b | aa=b, aabb=bb⟩φ(a) = b, φ(b) = bb
93054⟨a, b | aa=b, abab=bb⟩φ(a) = b, φ(b) = bb
93057⟨a, b | aa=b, abba=bb⟩φ(a) = b, φ(b) = bb
93064⟨a, b | aa=b, baab=bb⟩φ(a) = b, φ(b) = bb
93168⟨a, b | aa=b, bbb=aab⟩φ(a) = b, φ(b) = bb
93169⟨a, b | aa=b, bbb=aba⟩φ(a) = b, φ(b) = bb
106564⟨a, b | aaa=b, aaaa=bb⟩φ(a) = b, φ(b) = bbb
106566⟨a, b | aaa=b, aaab=ab⟩φ(a) = b, φ(b) = bbb
106567⟨a, b | aaa=b, aaab=ba⟩φ(a) = b, φ(b) = bbb
106570⟨a, b | aaa=b, aaba=ab⟩φ(a) = b, φ(b) = bbb
106571⟨a, b | aaa=b, aaba=ba⟩φ(a) = b, φ(b) = bbb
106663⟨a, b | aab=b, aaaa=ab⟩φ(a) = b, φ(b) = bbbbb
106763⟨a, b | aba=b, aaaa=ab⟩φ(a) = b, φ(b) = bbbbb
108918⟨a, b | aa=b, aaaab=bb⟩φ(a) = b, φ(b) = bb
108922⟨a, b | aa=b, aaaba=bb⟩φ(a) = b, φ(b) = bb
108929⟨a, b | aa=b, aabaa=bb⟩φ(a) = b, φ(b) = bb
109189⟨a, b | aa=b, aaaa=bbb⟩φ(a) = b, φ(b) = bb
109207⟨a, b | aa=b, aabb=aab⟩φ(a) = b, φ(b) = bb
109208⟨a, b | aa=b, aabb=aba⟩φ(a) = b, φ(b) = bb
109210⟨a, b | aa=b, aabb=baa⟩φ(a) = b, φ(b) = bb
109215⟨a, b | aa=b, abab=aab⟩φ(a) = b, φ(b) = bb
109216⟨a, b | aa=b, abab=aba⟩φ(a) = b, φ(b) = bb
109218⟨a, b | aa=b, abab=baa⟩φ(a) = b, φ(b) = bb
109223⟨a, b | aa=b, abba=aab⟩φ(a) = b, φ(b) = bb
109224⟨a, b | aa=b, abba=aba⟩φ(a) = b, φ(b) = bb
109237⟨a, b | aa=b, baab=aab⟩φ(a) = b, φ(b) = bb
109238⟨a, b | aa=b, baab=aba⟩φ(a) = b, φ(b) = bb
1120049⟨a, b | aab=b, aaaa=aab⟩φ(a) = b, φ(b) = bbbb
1120249⟨a, b | aba=b, aaaa=aab⟩φ(a) = b, φ(b) = bbbb
1120250⟨a, b | aba=b, aaaa=aba⟩φ(a) = b, φ(b) = bbbb
1124726⟨a, b | aa=b, aaaaaa=bb⟩φ(a) = b, φ(b) = bb
1125267⟨a, b | aa=b, aaaab=aab⟩φ(a) = b, φ(b) = bb
1125268⟨a, b | aa=b, aaaab=aba⟩φ(a) = b, φ(b) = bb
1125270⟨a, b | aa=b, aaaab=baa⟩φ(a) = b, φ(b) = bb
1125275⟨a, b | aa=b, aaaba=aab⟩φ(a) = b, φ(b) = bb
1125276⟨a, b | aa=b, aaaba=aba⟩φ(a) = b, φ(b) = bb
1125278⟨a, b | aa=b, aaaba=baa⟩φ(a) = b, φ(b) = bb
1125291⟨a, b | aa=b, aabaa=aab⟩φ(a) = b, φ(b) = bb
1125292⟨a, b | aa=b, aabaa=aba⟩φ(a) = b, φ(b) = bb
1125729⟨a, b | aa=b, aabb=aaaa⟩φ(a) = b, φ(b) = bb
1125735⟨a, b | aa=b, abab=aaaa⟩φ(a) = b, φ(b) = bb
1125740⟨a, b | aa=b, abba=aaaa⟩φ(a) = b, φ(b) = bb
1125756⟨a, b | aa=b, baab=aaaa⟩φ(a) = b, φ(b) = bb