#16343 ⟨a, b | aba=aa, aaab=bb⟩

Up:Monoids with two generators and two relations
Prev:#16342 ⟨a, b | aba=aa, aaab=ba⟩
Next:#16348 ⟨a, b | aba=aa, aabb=aa⟩

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.

1aba2abbaa3a2bba2baba4a3bba3ba2ba5a4ba6a5ba7a6b
11aba2abbaa3a2bba2baba4a3bba3ba2ba5a4ba6a5ba7a6b
aaa2aba3a2ba2a4a3ba3a2ba5a4ba4a3ba6a5ba7a6ba5a4b
bbbaa3bba2baba4ba3ba2ba5a4ba7a6ba6a5ba5a4ba6a5ba7a6b
a2a2a3a2ba4a3ba3a5a4ba4a3ba6a5ba5a4ba7a6ba5a4ba6a5b
ababa2a4ba3a2ba5a4a3ba6a5ba5a4ba7a6ba6a5ba7a6ba5a4b
bababa2babba3ba2bba2a7a6bba3ba2ba5a4ba7a6ba6a5ba7a6ba5a4b
a3a3a4a3ba5a4ba4a6a5ba5a4ba7a6ba6a5ba5a4ba6a5ba7a6b
a2ba2ba3a5ba4a3ba6a5a4ba7a6ba6a5ba5a4ba7a6ba5a4ba6a5b
ba2ba2ba3ba2ba7a6bba3a5a4ba7a6ba6a5ba5a4ba7a6ba5a4ba6a5b
babbabba2a4bba3ba2ba5a7a6ba6a5ba5a4ba7a6ba6a5ba7a6ba5a4b
a4a4a5a4ba6a5ba5a7a6ba6a5ba5a4ba7a6ba6a5ba7a6ba5a4b
a3ba3ba4a6ba5a4ba7a6a5ba5a4ba7a6ba6a5ba5a4ba6a5ba7a6b
ba3ba3a7a6ba5a4ba7a6a5ba5a4ba7a6ba6a5ba5a4ba6a5ba7a6b
ba2bba2bba3a5ba7a6ba6a5a4ba7a6ba6a5ba5a4ba7a6ba5a4ba6a5b
a5a5a6a5ba7a6ba6a5a4ba7a6ba6a5ba5a4ba7a6ba5a4ba6a5b
a4ba4ba5a4ba6a5ba5a7a6ba6a5ba5a4ba7a6ba6a5ba7a6ba5a4b
a6a6a7a6ba5a4ba7a6a5ba5a4ba7a6ba6a5ba5a4ba6a5ba7a6b
a5ba5ba6a5ba7a6ba6a5a4ba7a6ba6a5ba5a4ba7a6ba5a4ba6a5b
a7a7a5a4ba6a5ba5a7a6ba6a5ba5a4ba7a6ba6a5ba7a6ba5a4b
a6ba6ba7a6ba5a4ba7a6a5ba5a4ba7a6ba6a5ba5a4ba6a5ba7a6b

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. a8 ⇒ a5 [6]
2. ba4 ⇒ a7 [5]
3. aba ⇒ a2 [1]
4. a7b ⇒ a4b [4]
5. b2 ⇒ a3b [2]
6. ba3b ⇒ a6b [3]
# ab:aba=aa,aaab=bb a/b
aaaaaaaa=aaaaa
baaaa=aaaaaaa
aba=aa
aaaaaaab=aaaab
bb=aaab
baaab=aaaaaab

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
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