#2220 ⟨a, b | bb=aa, aaab=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.

1aba2aba3a2ba4
11aba2aba3a2ba4
aaa2aba3a2ba4aab
bbaba2a2ba3aa4a2
a2a2a3a2ba4aaba2a2b
ababa2ba3aa4a2aba3
a3a3a4aaba2a2ba3a
a2ba2baa4a2aba3a2ba4
a4a4aba2a2ba3aa4a2

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. ba ⇒ ab [7]
2. b2 ⇒ a2 [1]
3. a3b ⇒ a [2]
4. a5 ⇒ ab [4]
# ab:bb=aa,aaab=a ab - -
ba=ab
bb=aa
aaab=a
aaaaa=ab

Same cardinality

33 unique, 1189 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
91605⟨a, b | aaa=bb, abb=b⟩Noc Com 8ℕ(8 = 3)55
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
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.

22 total

Σ#PresentationMapping
92221⟨a, b | bb=aa, aaab=b⟩φ(a) = b, φ(b) = a
92222⟨a, b | bb=aa, aaba=a⟩φ(a) = a, φ(b) = b
92223⟨a, b | bb=aa, aaba=b⟩φ(a) = b, φ(b) = a
105204⟨a, b | aab=bb, abbb=a⟩φ(a) = ab, φ(b) = b
105212⟨a, b | aab=bb, babb=a⟩φ(a) = ab, φ(b) = b
105216⟨a, b | aab=bb, bbab=a⟩φ(a) = ab, φ(b) = b
105290⟨a, b | aba=bb, babb=a⟩φ(a) = ab, φ(b) = b
1112431⟨a, b | aabb=aa, abaa=b⟩φ(a) = b, φ(b) = aaa
1112486⟨a, b | aabb=bb, babb=a⟩φ(a) = aaa, φ(b) = b
1112546⟨a, b | abab=bb, abbb=a⟩φ(a) = aaa, φ(b) = b
1112548⟨a, b | abab=bb, babb=a⟩φ(a) = aaa, φ(b) = b
1112550⟨a, b | abab=bb, bbab=a⟩φ(a) = aaa, φ(b) = b
1112592⟨a, b | abba=bb, abbb=a⟩φ(a) = aaa, φ(b) = b
1112600⟨a, b | abba=bb, babb=a⟩φ(a) = aaa, φ(b) = b
1112624⟨a, b | baab=bb, babb=a⟩φ(a) = aaa, φ(b) = b
1115743⟨a, b | aab=bb, aaabb=a⟩φ(a) = ab, φ(b) = b
1115747⟨a, b | aab=bb, aabab=a⟩φ(a) = ab, φ(b) = b
1115755⟨a, b | aab=bb, abaab=a⟩φ(a) = ab, φ(b) = b
1115771⟨a, b | aab=bb, baaab=a⟩φ(a) = ab, φ(b) = b
1115911⟨a, b | aba=bb, aaabb=a⟩φ(a) = ab, φ(b) = b
1115917⟨a, b | aba=bb, aabba=a⟩φ(a) = ab, φ(b) = b
1115921⟨a, b | aba=bb, abaab=a⟩φ(a) = ab, φ(b) = b