#2220 ⟨a, b | bb=aa, aaab=a⟩

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Quick links

  1. Properties
  2. Elements
  3. Staircase diagram
  4. Cayley table
  5. Right Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Elements

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.

1aba2a3a4a5a6
11aba2a3a4a5a6
aaa2a5a3a4a5a6a
bba5a2a6aa2a3a4
a2a2a3a6a4a5a6aa2
a3a3a4aa5a6aa2a3
a4a4a5a2a6aa2a3a4
a5a5a6a3aa2a3a4a5
a6a6aa4a2a3a4a5a6

Right Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:
Word to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 1 
1
#RuleProof
1. a7 ⇒ a [7]
2. ba ⇒ a5 [8]
3. ab ⇒ a5 [5]
4. b2 ⇒ a2 [1]
# ab:bb=aa,aaab=a a/b
aaaaaaa=a
ba=aaaaa
ab=aaaaa
bb=aa

Same cardinality

33 unique, 1189 total

Σ#PresentationDescriptionRelated
7188⟨a, b | aab=1, bbbb=1⟩Isomorphic to ℤ8727 iso
7330⟨a, b | aa=1, abba=b⟩Finite non-commutative monoid with 8 elements60 iso
8556⟨a, b | aba=b, aabb=1⟩Finite non-Abelian group with 8 elements28 iso
8898⟨a, b | aa=a, bbbb=a⟩Isomorphic to ℕ(8 = 4)48 iso
8918⟨a, b | aa=b, bbbb=a⟩Isomorphic to ℕ(8 = 1)34 iso
8919⟨a, b | aa=b, bbbb=b⟩Isomorphic to ℕ(8 = 2)46 iso
8961⟨a, b | aa=a, aba=bb⟩Finite non-commutative monoid with 8 elements5 iso
91605⟨a, b | aaa=bb, abb=b⟩Isomorphic to ℕ(8 = 3)55 iso
91606⟨a, b | aaa=bb, bab=a⟩Finite commutative monoid with 8 elements14 iso
91615⟨a, b | aab=aa, baa=b⟩Finite non-commutative monoid with 8 elements9 iso, 5 anti-iso
91650⟨a, b | aab=bb, baa=a⟩Finite non-commutative monoid with 8 elements4 iso, 11 anti-iso
92206⟨a, b | ab=aa, bbbb=a⟩Isomorphic to ℕ(8 = 5)32 iso
92247⟨a, b | ab=aa, baa=bb⟩Finite non-commutative monoid with 8 elements1 iso
92256⟨a, b | ab=aa, bbb=aa⟩Finite non-commutative monoid with 8 elements1 iso
92258⟨a, b | ab=aa, bbb=ba⟩Finite non-commutative monoid with 8 elements
92883⟨a, b | aa=a, abbbb=b⟩Finite non-commutative monoid with 8 elements14 iso
93107⟨a, b | ab=a, baaa=bb⟩Finite non-commutative monoid with 8 elements9 iso
93123⟨a, b | ab=a, bbaa=bb⟩Finite non-commutative monoid with 8 elements10 iso
105191⟨a, b | aab=bb, aaaa=b⟩Isomorphic to ℕ(8 = 6)18 iso
106664⟨a, b | aab=b, aaaa=ba⟩Finite non-commutative monoid with 8 elements1 iso
107057⟨a, b | ab=aa, aaaa=bb⟩Finite non-commutative monoid with 8 elements7 iso
109380⟨a, b | ab=a, bbbb=baa⟩Finite non-commutative monoid with 8 elements2 iso
1112606⟨a, b | abbb=aa, bbbb=a⟩Isomorphic to ℕ(8 = 7)14 iso
1120047⟨a, b | aab=a, bbbb=bbb⟩Finite non-commutative monoid with 8 elements
1120051⟨a, b | aab=b, aaaa=abb⟩Finite commutative monoid with 8 elements1 iso
1124863⟨a, b | ab=a, aaaaaa=bb⟩Finite commutative monoid with 8 elements
1125112⟨a, b | ab=a, bbbbbb=aa⟩Finite commutative monoid with 8 elements
1125114⟨a, b | ab=a, bbbbbb=ba⟩Finite non-commutative monoid with 8 elements
1125411⟨a, b | ab=a, aaaaa=bbb⟩Finite commutative monoid with 8 elements
1125652⟨a, b | ab=a, bbbbb=aaa⟩Finite commutative monoid with 8 elements
1125658⟨a, b | ab=a, bbbbb=bba⟩Finite non-commutative monoid with 8 elements
1125897⟨a, b | ab=a, bbbb=aaaa⟩Finite commutative monoid with 8 elements
1125911⟨a, b | ab=a, bbbb=bbba⟩Finite non-commutative monoid with 8 elements

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) = aaaaa, φ(b) = b
105212⟨a, b | aab=bb, babb=a⟩φ(a) = aaaaa, φ(b) = b
105216⟨a, b | aab=bb, bbab=a⟩φ(a) = aaaaa, φ(b) = b
105290⟨a, b | aba=bb, babb=a⟩φ(a) = aaaaa, 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) = aaaaa, φ(b) = b
1115747⟨a, b | aab=bb, aabab=a⟩φ(a) = aaaaa, φ(b) = b
1115755⟨a, b | aab=bb, abaab=a⟩φ(a) = aaaaa, φ(b) = b
1115771⟨a, b | aab=bb, baaab=a⟩φ(a) = aaaaa, φ(b) = b
1115911⟨a, b | aba=bb, aaabb=a⟩φ(a) = aaaaa, φ(b) = b
1115917⟨a, b | aba=bb, aabba=a⟩φ(a) = aaaaa, φ(b) = b
1115921⟨a, b | aba=bb, abaab=a⟩φ(a) = aaaaa, φ(b) = b