#5349 ⟨a, b | aaa=bb, bab=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
  8. Isomorphic instances

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.

1aba2abbaa3ababa2a4ba3a5ba4a6ba5a7ba6a8ba7a9
11aba2abbaa3ababa2a4ba3a5ba4a6ba5a7ba6a8ba7a9
aaa2aba3ba4abaa4ba5ba7a5ba2a6ba3a7ba4a8ba5a9ba6a4
bbbaa3ba2a2a4ba3a3a5ba4a6ba5a7ba6a8ba7a9ba2a4ba3
a2a2a3ba4a4ba3ba5a5ba4ba6a6ba7a7ba2a8ba3a9ba4a4ba5a5
abababaa4ba7a3a5ba2a4a6ba3a7ba4a8ba5a9ba6a4ba7a5ba2
bababa2a2ba3a7a3ba4a8a4ba5a5ba6a6ba7a7ba2a8ba3a9ba4
a3a3a4ba3a5ba2ba4a6ba3ba5a7ba6a8ba7a9ba2a4ba3a5ba4a6
abaababa7a3ba2a8a4ba3a9a5ba4a6ba5a7ba6a8ba7a9ba2a4ba3
ba2ba2ba3a7ba4a6a8ba5a7a9ba6a4ba7a5ba2a6ba3a7ba4a8ba5
a4a4a5ba2a6ba7ba3a7ba2ba4a8ba5a9ba6a4ba7a5ba2a6ba3a7
ba3ba3ba4a6ba5a5a7ba6a6a8ba7a9ba2a4ba3a5ba4a6ba5a7ba6
a5a5a6ba7a7ba6ba2a8ba7ba3a9ba4a4ba5a5ba6a6ba7a7ba2a8
ba4ba4ba5a5ba6a4a6ba7a5a7ba2a8ba3a9ba4a4ba5a5ba6a6ba7
a6a6a7ba6a8ba5ba7a9ba6ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9
ba5ba5ba6a4ba7a9a5ba2a4a6ba3a7ba4a8ba5a9ba6a4ba7a5ba2
a7a7a8ba5a9ba4ba6a4ba5ba7a5ba2a6ba3a7ba4a8ba5a9ba6a4
ba6ba6ba7a9ba2a8a4ba3a9a5ba4a6ba5a7ba6a8ba7a9ba2a4ba3
a8a8a9ba4a4ba3ba5a5ba4ba6a6ba7a7ba2a8ba3a9ba4a4ba5a5
ba7ba7ba2a8ba3a7a9ba4a8a4ba5a5ba6a6ba7a7ba2a8ba3a9ba4
a9a9a4ba3a5ba2ba4a6ba3ba5a7ba6a8ba7a9ba2a4ba3a5ba4a6

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. a10 ⇒ a4 [6]
2. ba8 ⇒ ba2 [8]
3. aba2 ⇒ ba7 [7]
4. a2b ⇒ ba4 [5]
5. b2 ⇒ a3 [1]
6. bab ⇒ a2 [2]
# ab:aaa=bb,bab=aa a/b
aaaaaaaaaa=aaaa
baaaaaaaa=baa
abaa=baaaaaaa
aab=baaaa
bb=aaa
bab=aa

Same cardinality

20 unique, 64 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
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
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

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

1 total

Σ#PresentationMapping
1116452⟨a, b | aba=bb, abab=aa⟩φ(a) = b, φ(b) = a