#21047 ⟨a, b | ab=aa, bbbb=bbb⟩

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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.
A left zero element x satisfies xy = x for all y.
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.

1aba2bab2a3ba2b2ab3a4ba3b2a2b3aba4b2a3b3a2b2a4b3a3b3a4
11aba2bab2a3ba2b2ab3a4ba3b2a2b3aba4b2a3b3a2b2a4b3a3b3a4
aaa2a2a3a3a3a4a4a4a4a4a4a4a4a4a4a4a4a4a4
bbbab2ba2b2ab3ba3b2a2b3ab3ba4b2a3b3a2b3ab2a4b3a3b3a2b3a4b3a3b3a4
a2a2a3a3a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4
bababa2ba2ba3ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2b2b2ab3b2a2b3ab3b2a3b3a2b3ab3b2a4b3a3b3a2b3ab3a4b3a3b3a2b3a4b3a3b3a4
a3a3a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4
ba2ba2ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2ab2ab2a2b2a2b2a3b2a3b2a3b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4
b3b3b3ab3b3a2b3ab3b3a3b3a2b3ab3b3a4b3a3b3a2b3ab3a4b3a3b3a2b3a4b3a3b3a4
a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4
ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2a2b2a2b2a3b2a3b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4
b3ab3ab3a2b3a2b3a3b3a3b3a3b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4
ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2a3b2a3b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4
b3a2b3a2b3a3b3a3b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4
b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4
b3a3b3a3b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4
b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4b3a4

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. ab ⇒ a2 [1]
2. b4 ⇒ b3 [2]
3. a5 ⇒ a4 [3]
# ab:ab=aa,bbbb=bbb ab
ab=aa
bbbb=bbb
aaaaa=aaaa

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
1119503⟨a, b | aab=a, bbbbb=bb⟩Finite non-commutative monoid with 20 elements