#19503 ⟨a, b | aab=a, bbbbb=bb⟩

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Quick links

  1. Properties
  2. Elements
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality

Properties

Elements

Elements in the center commute with all other 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.

Cayley table

Idempotents are shown in bold.

1aba2bab2a3ba2b2ab3ba3b2a2b3ab4b2a3b3a2b4ab3a3b4a2b4a3
11aba2bab2a3ba2b2ab3ba3b2a2b3ab4b2a3b3a2b4ab3a3b4a2b4a3
aaa2a3a3aa2aa2a3aa3aa2a3a2a3aaa2a3
bbbab2ba2b2ab3ba3b2a2b3ab4b2a3b3a2b4ab2b3a3b4a2b2ab4a3b2a2b2a3
a2a2a3aaa2a3a2a3aa2aa2a3aa3aa2a2a3a
bababa2ba3ba3baba2baba2ba3baba3baba2ba3ba2ba3bababa2ba3
b2b2b2ab3b2a2b3ab4b2a3b3a2b4ab2b3a3b4a2b2ab3b4a3b2a2b3ab2a3b3a2b3a3
a3a3aa2a2a3aa3aa2a3a2a3aa2aa2a3a3aa2
ba2ba2ba3bababa2ba3ba2ba3baba2baba2ba3baba3baba2ba2ba3ba
b2ab2ab2a2b2a3b2a3b2ab2a2b2ab2a2b2a3b2ab2a3b2ab2a2b2a3b2a2b2a3b2ab2ab2a2b2a3
b3b3b3ab4b3a2b4ab2b3a3b4a2b2ab3b4a3b2a2b3ab4b2a3b3a2b4ab3a3b4a2b4a3
ba3ba3baba2ba2ba3baba3baba2ba3ba2ba3baba2baba2ba3ba3baba2
b2a2b2a2b2a3b2ab2ab2a2b2a3b2a2b2a3b2ab2a2b2ab2a2b2a3b2ab2a3b2ab2a2b2a2b2a3b2a
b3ab3ab3a2b3a3b3a3b3ab3a2b3ab3a2b3a3b3ab3a3b3ab3a2b3a3b3a2b3a3b3ab3ab3a2b3a3
b4b4b4ab2b4a2b2ab3b4a3b2a2b3ab4b2a3b3a2b4ab2b3a3b4a2b2ab4a3b2a2b2a3
b2a3b2a3b2ab2a2b2a2b2a3b2ab2a3b2ab2a2b2a3b2a2b2a3b2ab2a2b2ab2a2b2a3b2a3b2ab2a2
b3a2b3a2b3a3b3ab3ab3a2b3a3b3a2b3a3b3ab3a2b3ab3a2b3a3b3ab3a3b3ab3a2b3a2b3a3b3a
b4ab4ab4a2b4a3b4a3b4ab4a2b4ab4a2b4a3b4ab4a3b4ab4a2b4a3b4a2b4a3b4ab4ab4a2b4a3
b3a3b3a3b3ab3a2b3a2b3a3b3ab3a3b3ab3a2b3a3b3a2b3a3b3ab3a2b3ab3a2b3a3b3a3b3ab3a2
b4a2b4a2b4a3b4ab4ab4a2b4a3b4a2b4a3b4ab4a2b4ab4a2b4a3b4ab4a3b4ab4a2b4a2b4a3b4a
b4a3b4a3b4ab4a2b4a2b4a3b4ab4a3b4ab4a2b4a3b4a2b4a3b4ab4a2b4ab4a2b4a3b4a3b4ab4a2

Right Cayley graph

Idempotents are shown in bold.

Left 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 [7]
2. ab ⇒ a3 [6]
3. b5 ⇒ b2 [2]
# ab:aab=a,bbbbb=bb a/b
aaaa=a
ab=aaa
bbbbb=bb

Same cardinality

20 unique, 65 total

Σ#PresentationDescriptionRelated
8526⟨a, b | aab=a, bbbb=1⟩Finite non-commutative monoid with 20 elements10 iso, 9 anti-iso
92207⟨a, b | ab=aa, bbbb=b⟩Finite non-commutative monoid with 20 elements1 anti-iso
104095⟨a, b | baa=abb, abab=1⟩Finite non-Abelian group with 20 elements5 iso
104212⟨a, b | aaaaa=1, abbbb=1⟩Isomorphic to ℤ2016 iso
105349⟨a, b | aaa=bb, bab=aa⟩Finite non-commutative monoid with 20 elements1 iso
106728⟨a, b | aba=a, aaaa=bb⟩Finite non-commutative monoid with 20 elements
106764⟨a, b | aba=b, aaaa=bb⟩Finite non-commutative monoid with 20 elements
107117⟨a, b | ab=aa, bbbb=bb⟩Finite non-commutative monoid with 20 elements
1112159⟨a, b | aaaa=aa, abbb=b⟩Finite non-commutative monoid with 20 elements
1112322⟨a, b | aaab=bb, bbba=a⟩Finite non-commutative monoid with 20 elements
1114367⟨a, b | aaaa=a, bbbbb=a⟩Isomorphic to ℕ(20 = 5)
1114407⟨a, b | aaaa=b, bbbbb=a⟩Isomorphic to ℕ(20 = 1)
1114408⟨a, b | aaaa=b, bbbbb=b⟩Isomorphic to ℕ(20 = 4)
1115933⟨a, b | aba=bb, baaab=a⟩Finite non-commutative monoid with 20 elements
1116124⟨a, b | aab=aa, baba=bb⟩Finite non-commutative monoid with 20 elements
1116339⟨a, b | aba=aa, aaaa=bb⟩Finite non-commutative monoid with 20 elements
1116343⟨a, b | aba=aa, aaab=bb⟩Finite non-commutative monoid with 20 elements
1116545⟨a, b | aba=bb, abb=aaa⟩Finite non-commutative monoid with 20 elements
1118619⟨a, b | aba=b, aaaaabb=1⟩Finite non-Abelian group with 20 elements3 iso
1121047⟨a, b | ab=aa, bbbb=bbb⟩Finite non-commutative monoid with 20 elements