#16313 ⟨a, b | aab=bb, baba=aa⟩

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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.

1aba2abbaa3a2babababa4a3b(ab)2a5a4ba6a5ba6b
11aba2abbaa3a2babababa4a3b(ab)2a5a4ba6a5ba6b
aaa2aba3a2babaa4a3ba5(ab)2a5a4ba5ba6a5ba3a6ba3b
bbbaa2ba4baba5a5a4ba2a5ba6a5ba2ba3a6ba4a3ba4b
a2a2a3a2ba4a3ba5a5a4ba6a5ba6a5ba6ba3a6ba4a3ba4b
abababaa3ba5(ab)2a6a6a5ba3a6ba3a6ba3ba4a3ba5a4ba5b
babaa4baba5a4ba2a6a5ba3a2ba3a6ba3ba4a3ba5a4ba5b
a3a3a4a3ba5a4ba6a6a5ba3a6ba3a6ba3ba4a3ba5a4ba5b
a2ba2ba5a4ba6a5ba3a3a6ba4a3ba4a3ba4ba5a4ba6a5ba6b
abaabaa5(ab)2a6a5ba3a3a6ba4a3ba4a3ba4ba5a4ba6a5ba6b
babbaba2a5ba3a2ba4a4a3ba5a4ba5a4ba5ba6a5ba3a6ba3b
a4a4a5a4ba6a5ba3a3a6ba4a3ba4a3ba4ba5a4ba6a5ba6b
a3ba3ba6a5ba3a6ba4a4a3ba5a4ba5a4ba5ba6a5ba3a6ba3b
(ab)2(ab)2a3a6ba4a3ba5a5a4ba6a5ba6a5ba6ba3a6ba4a3ba4b
a5a5a6a5ba3a6ba4a4a3ba5a4ba5a4ba5ba6a5ba3a6ba3b
a4ba4ba3a6ba4a3ba5a5a4ba6a5ba6a5ba6ba3a6ba4a3ba4b
a6a6a3a6ba4a3ba5a5a4ba6a5ba6a5ba6ba3a6ba4a3ba4b
a5ba5ba4a3ba5a4ba6a6a5ba3a6ba3a6ba3ba4a3ba5a4ba5b
a6ba6ba5a4ba6a5ba3a3a6ba4a3ba4a3ba4ba5a4ba6a5ba6b

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. ba2 ⇒ a4 [3]
3. a2ba ⇒ a5 [4]
4. b2 ⇒ a2b [1]
5. (ba)2 ⇒ a2 [2]
# ab:aab=bb,baba=aa a/b
aaaaaaa=aaa
baa=aaaa
aaba=aaaaa
bb=aab
baba=aa

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
1116043⟨a, b | aaa=ab, bbbb=ba⟩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