#12322 ⟨a, b | aaab=bb, bbba=a⟩

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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.

1aba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8ba9b
11aba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8ba9b
aaa2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9bab
bba4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba3b
a2a2a3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba2b
ababa5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba4a3ba4b
a3a3a4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba3b
a2ba2ba6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba4a3ba5a4ba5b
a4a4a5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba4a3ba4b
a3ba3ba7a6ba8a7ba9a8baa9ba2aba3a2ba4a3ba5a4ba6a5ba6b
a5a5a6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba4a3ba5a4ba5b
a4ba4ba8a7ba9a8baa9ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba7b
a6a6a7a6ba8a7ba9a8baa9ba2aba3a2ba4a3ba5a4ba6a5ba6b
a5ba5ba9a8baa9ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba8b
a7a7a8a7ba9a8baa9ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba7b
a6ba6baa9ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8ba9b
a8a8a9a8baa9ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba8b
a7ba7ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9bab
a9a9aa9ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8ba9b
a8ba8ba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba2b
a9ba9ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba3b

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. a10 ⇒ a [8]
2. ba ⇒ a4 [7]
3. b2 ⇒ a3b [1]
# ab:aaab=bb,bbba=a a/b
aaaaaaaaaa=a
ba=aaaa
bb=aaab

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
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