#12159 ⟨a, b | aaaa=aa, abbb=b⟩

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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
aaa2b3a3b3ab4a2b3a2b4abb3a3b4a2bab2b4a3ba2b2aba3b2a2b2a3
bbbab2ba2b2ab3ba3b2a2b3ab4b2a3b3a2b4abb3a3b4a2bab4a3ba2ba3
a2a2a3ba2bab2a3ba2b2ab3ba3b2a2b3ab4b2a3b3a2b4ab3a3b4a2b4a3
bababa2b4ba3b4abba2b4a2bab2b4a3ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
b2b2b2ab3b2a2b3ab4b2a3b3a2b4abb3a3b4a2bab2b4a3ba2b2aba3b2a2b2a3
a3a3a2b3a3b3ab4a2b3a2b4abb3a3b4a2bab2b4a3ba2b2aba3b2a2b2a3
ba2ba2ba3b2ba2b2ab3ba3b2a2b3ab4b2a3b3a2b4abb3a3b4a2bab4a3ba2ba3
b2ab2ab2a2bb2a3bab2b2a2ba2b2ab3ba3b2a2b3ab4b2a3b3a2b4ab3a3b4a2b4a3
b3b3b3ab4b3a2b4abb3a3b4a2bab2b4a3ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
ba3ba3ba2b4ba3b4abba2b4a2bab2b4a3ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
b2a2b2a2b2a3b3b2a2b3ab4b2a3b3a2b4abb3a3b4a2bab2b4a3ba2b2aba3b2a2b2a3
b3ab3ab3a2b2b3a3b2ab3b3a2b2a2b3ab4b2a3b3a2b4abb3a3b4a2bab4a3ba2ba3
b4b4b4abb4a2bab2b4a3ba2b2ab3ba3b2a2b3ab4b2a3b3a2b4ab3a3b4a2b4a3
b2a3b2a3b2a2bb2a3bab2b2a2ba2b2ab3ba3b2a2b3ab4b2a3b3a2b4ab3a3b4a2b4a3
b3a2b3a2b3a3b4b3a2b4abb3a3b4a2bab2b4a3ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
b4ab4ab4a2b3b4a3b3ab4b4a2b3a2b4abb3a3b4a2bab2b4a3ba2b2aba3b2a2b2a3
b3a3b3a3b3a2b2b3a3b2ab3b3a2b2a2b3ab4b2a3b3a2b4abb3a3b4a2bab4a3ba2ba3
b4a2b4a2b4a3bb4a2bab2b4a3ba2b2ab3ba3b2a2b3ab4b2a3b3a2b4ab3a3b4a2b4a3
b4a3b4a3b4a2b3b4a3b3ab4b4a2b3a2b4abb3a3b4a2bab2b4a3ba2b2aba3b2a2b2a3

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. b5 ⇒ b [6]
2. ab ⇒ b3 [5]
3. a4 ⇒ a2 [1]
# ab:aaaa=aa,abbb=b b/a
bbbbb=b
ab=bbb
aaaa=aa

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
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
1119503⟨a, b | aab=a, bbbbb=bb⟩Finite non-commutative monoid with 20 elements
1121047⟨a, b | ab=aa, bbbb=bbb⟩Finite non-commutative monoid with 20 elements