#16043 ⟨a, b | aaa=ab, bbbb=ba⟩

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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.

1aba2bab2a3ba2b3a4ba3a5ba4a6ba5a7ba6a8
11aba2bab2a3ba2b3a4ba3a5ba4a6ba5a7ba6a8
aaa2a3a3a4a5a4a5a7a5a6a6a7a7a8a8a4a4
bbbab2ba2ba3b3ba3ba4baba4ba5ba5ba6ba6ba2ba2ba3ba3
a2a2a3a4a4a5a6a5a6a8a6a7a7a8a8a4a4a5a5
bababa2ba3ba3ba4ba5ba4ba5ba2ba5ba6ba6ba2ba2ba3ba3ba4ba4
b2b2ba3b3ba4ba5baba5ba6ba3ba6ba2ba2ba3ba3ba4ba4ba5ba5
a3a3a4a5a5a6a7a6a7a4a7a8a8a4a4a5a5a6a6
ba2ba2ba3ba4ba4ba5ba6ba5ba6ba3ba6ba2ba2ba3ba3ba4ba4ba5ba5
b3b3ba5baba6ba2ba3ba2ba3ba5ba3ba4ba4ba5ba5ba6ba6ba2ba2
a4a4a5a6a6a7a8a7a8a5a8a4a4a5a5a6a6a7a7
ba3ba3ba4ba5ba5ba6ba2ba6ba2ba4ba2ba3ba3ba4ba4ba5ba5ba6ba6
a5a5a6a7a7a8a4a8a4a6a4a5a5a6a6a7a7a8a8
ba4ba4ba5ba6ba6ba2ba3ba2ba3ba5ba3ba4ba4ba5ba5ba6ba6ba2ba2
a6a6a7a8a8a4a5a4a5a7a5a6a6a7a7a8a8a4a4
ba5ba5ba6ba2ba2ba3ba4ba3ba4ba6ba4ba5ba5ba6ba6ba2ba2ba3ba3
a7a7a8a4a4a5a6a5a6a8a6a7a7a8a8a4a4a5a5
ba6ba6ba2ba3ba3ba4ba5ba4ba5ba2ba5ba6ba6ba2ba2ba3ba3ba4ba4
a8a8a4a5a5a6a7a6a7a4a7a8a8a4a4a5a5a6a6

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. a9 ⇒ a4 [3]
2. ba7 ⇒ ba2 [5]
3. ab ⇒ a3 [1]
4. b2a ⇒ ba3 [4]
5. b4 ⇒ ba [2]
# ab:aaa=ab,bbbb=ba a/b
aaaaaaaaa=aaaa
baaaaaaa=baa
ab=aaa
bba=baaa
bbbb=ba

Same cardinality

17 unique, 104 total

Σ#PresentationDescriptionRelated
81121⟨a, b | aa=1, abbba=b⟩Finite non-commutative monoid with 18 elements17 iso
93387⟨a, b | aa=1, abbbbb=b⟩Finite non-commutative monoid with 18 elements22 iso, 9 anti-iso
105521⟨a, b | aaab=1, bbbbbb=1⟩Isomorphic to ℤ1833 iso
106732⟨a, b | aba=a, aaab=bb⟩Finite non-commutative monoid with 18 elements
106788⟨a, b | aba=b, baab=aa⟩Finite non-commutative monoid with 18 elements
106795⟨a, b | aba=b, bbbb=aa⟩Finite non-commutative monoid with 18 elements
107039⟨a, b | bb=aa, aaaba=b⟩Finite non-commutative monoid with 18 elements2 iso
108910⟨a, b | aa=a, bbbbb=ab⟩Finite non-commutative monoid with 18 elements1 iso
1112187⟨a, b | aaaa=ab, babb=b⟩Finite non-commutative monoid with 18 elements2 iso
1115797⟨a, b | aab=bb, bbbba=a⟩Finite non-commutative monoid with 18 elements
1116042⟨a, b | aaa=ab, bbbb=ab⟩Finite non-commutative monoid with 18 elements
1116313⟨a, b | aab=bb, baba=aa⟩Finite non-commutative monoid with 18 elements
1118758⟨a, b | aaa=a, bbbbbb=a⟩Isomorphic to ℕ(18 = 6)
1118830⟨a, b | aaa=b, bbbbbb=a⟩Isomorphic to ℕ(18 = 1)
1118831⟨a, b | aaa=b, bbbbbb=b⟩Isomorphic to ℕ(18 = 3)
1119624⟨a, b | aab=b, bbbba=aa⟩Finite non-commutative monoid with 18 elements
1120914⟨a, b | bb=aa, ababa=aa⟩Finite non-commutative monoid with 18 elements1 iso