#6728 ⟨a, b | aba=a, aaaa=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.

1aba2abbab2a3a2bba2babb3a3bba3ba2bb4ba3bb5b6b7
11aba2abbab2a3a2bba2babb3a3bba3ba2bb4ba3bb5b6b7
aaa2aba3a2bab7b2a3ba2abb2b3a3a2bb3a3bb4b5b6
bbbab2ba2babb7b3ba3ba2bb6b2b4ba3bb5b7b5b6b6b7b2
a2a2a3a2bb2a3ba2b6b7b3a3a2bb7b2b2a3bb2b3b3b4b5
ababab7a2abb6b2a3a2bb5b7b3a3bb4b6b4b5b5b6b7
bababa2babba3ba2bbab2b3ba3bba2babb3b4ba3ba2bb4ba3bb5b6b7
b2b2b7b3b6b2b2b4b5b7b7b3b5b6b6b2b6b7b7b2b3
a3a3b2a3bb7b3a3b5b6b2b2a3bb6b7b7b3b7b2b2b3b4
a2ba2ba2b6a3a2bb5b7b2a3bb4b6b2b3b3b5b3b4b4b5b6
ba2ba2ba3ba2bb3ba3bba2b7b2b4ba3ba2bb2b3b3ba3bb3b4b4b5b6
babbabbab2ba2babb7b3ba3ba2bb6b2b4ba3bb5b7b5b6b6b7b2
b3b3b2b4b7b3b3b5b6b2b2b4b6b7b7b3b7b2b2b3b4
a3ba3ba3b5b2a3bb4b6b7b3b3b5b7b2b2b4b2b3b3b4b5
ba3ba3b3ba3bb2b4ba3b6b7b3b3ba3bb7b2b2b4b2b3b3b4b5
ba2bba2bba2b7ba3ba2bb6b2b3ba3bb5b7b3b4b4b6b4b5b5b6b7
b4b4b3b5b2b4b4b6b7b3b3b5b7b2b2b4b2b3b3b4b5
ba3bba3bba3b6b3ba3bb5b7b2b4b4b6b2b3b3b5b3b4b4b5b6
b5b5b4b6b3b5b5b7b2b4b4b6b2b3b3b5b3b4b4b5b6
b6b6b5b7b4b6b6b2b3b5b5b7b3b4b4b6b4b5b5b6b7
b7b7b6b2b5b7b7b3b4b6b6b2b4b5b5b7b5b6b6b7b2

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. b8 ⇒ b2 [10]
2. ab2 ⇒ b7 [11]
3. b2a ⇒ b7 [12]
4. aba ⇒ a [1]
5. a4 ⇒ b2 [2]
# ab:aba=a,aaaa=bb b/a
bbbbbbbb=bb
abb=bbbbbbb
bba=bbbbbbb
aba=a
aaaa=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
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
1121047⟨a, b | ab=aa, bbbb=bbb⟩Finite non-commutative monoid with 20 elements