#6732 ⟨a, b | aba=a, aaab=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.

1aba2abbaa3a2bba2baba4a3bba2ba5a4ba6a5ba6b
11aba2abbaa3a2bba2baba4a3bba2ba5a4ba6a5ba6b
aaa2aba3a2baa4a3ba2aba5a4ba2ba6a5ba3a6ba3b
bbbaa3bba2baba3a6ba2ba4a3ba3a6ba4ba4a3ba5a4ba5b
a2a2a3a2ba4a3ba2a5a4ba3a2ba6a5ba3ba3a6ba4a3ba4b
ababaa4ba2aba4a3a2ba5a4ba4a3ba5ba5a4ba6a5ba6b
bababa2baba6ba2bbaa3a6bba2baba4a3bba2ba5a4ba6a5ba6b
a3a3a4a3ba5a4ba3a6a5ba4a3ba3a6ba4ba4a3ba5a4ba5b
a2ba2ba2a5ba3a2ba5a4a3ba6a5ba5a4ba6ba6a5ba3a6ba3b
ba2ba2a6ba2ba3a6bba2a4a3ba6ba2ba5a4ba6ba6a5ba3a6ba3b
babbabbaa3bba2baba3a6ba2ba4a3ba3a6ba4ba4a3ba5a4ba5b
a4a4a5a4ba6a5ba4a3a6ba5a4ba4a3ba5ba5a4ba6a5ba6b
a3ba3ba3a6ba4a3ba6a5a4ba3a6ba6a5ba3ba3a6ba4a3ba4b
ba2bba2bba2a4ba6ba2ba4a3a6ba5a4ba4a3ba5ba5a4ba6a5ba6b
a5a5a6a5ba3a6ba5a4a3ba6a5ba5a4ba6ba6a5ba3a6ba3b
a4ba4ba4a3ba5a4ba3a6a5ba4a3ba3a6ba4ba4a3ba5a4ba5b
a6a6a3a6ba4a3ba6a5a4ba3a6ba6a5ba3ba3a6ba4a3ba4b
a5ba5ba5a4ba6a5ba4a3a6ba5a4ba4a3ba5ba5a4ba6a5ba6b
a6ba6ba6a5ba3a6ba5a4a3ba6a5ba5a4ba6ba6a5ba3a6ba3b

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. a7 ⇒ a3 [5]
2. ba3 ⇒ a6 [4]
3. aba ⇒ a [1]
4. b2 ⇒ a3b [2]
# ab:aba=a,aaab=bb a/b
aaaaaaa=aaa
baaa=aaaaaa
aba=a
bb=aaab

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
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
1116043⟨a, b | aaa=ab, bbbb=ba⟩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