#1606 ⟨a, b | aaa=bb, bab=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.

1aba2baa3ba2ba3
11aba2baa3ba2ba3
aaa2baa3ba2aba3ba
bbbaa3ba2aba3a2a3
a2a2a3ba2aba3a2baba2
bababa2aba3a2baa3a
a3a3aba3a2baa3ba2ba3
ba2ba2ba3a2baa3ba2aa2
ba3ba3baa3ba2aba3a2a3

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. a4 ⇒ a [6]
2. ab ⇒ ba [5]
3. b2 ⇒ a3 [1]
# ab:aaa=bb,bab=a a/b
aaaa=a
ab=ba
bb=aaa

Same cardinality

33 unique, 1197 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
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
92220⟨a, b | bb=aa, aaab=a⟩Finite commutative monoid with 8 elements22 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.

14 total

Σ#PresentationMapping
106690⟨a, b | aab=b, abbb=aa⟩φ(a) = b, φ(b) = ba
106706⟨a, b | aab=b, babb=aa⟩φ(a) = b, φ(b) = ba
106714⟨a, b | aab=b, bbab=aa⟩φ(a) = b, φ(b) = ba
106784⟨a, b | aba=b, abbb=aa⟩φ(a) = b, φ(b) = ba
106791⟨a, b | aba=b, babb=aa⟩φ(a) = b, φ(b) = ba
1119548⟨a, b | aab=b, ababb=aa⟩φ(a) = b, φ(b) = a
1119556⟨a, b | aab=b, abbab=aa⟩φ(a) = b, φ(b) = a
1119588⟨a, b | aab=b, babab=aa⟩φ(a) = b, φ(b) = a
1119730⟨a, b | aba=b, aabbb=aa⟩φ(a) = b, φ(b) = a
1119741⟨a, b | aba=b, ababb=aa⟩φ(a) = b, φ(b) = a
1119745⟨a, b | aba=b, abbab=aa⟩φ(a) = b, φ(b) = a
1119749⟨a, b | aba=b, abbba=aa⟩φ(a) = b, φ(b) = a
1119759⟨a, b | aba=b, baabb=aa⟩φ(a) = b, φ(b) = a
1119763⟨a, b | aba=b, babab=aa⟩φ(a) = b, φ(b) = a