#13151 ⟨a, b | aba=aaa, baa=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.
A left zero element x satisfies xy = x for all y.
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.

1aba2abbaa3a2bba2baba4a3bba3a5ba4ba3ba6
11aba2abbaa3a2bba2baba4a3bba3a5ba4ba3ba6
aaa2aba3a2ba3a4a3ba4a3ba5a6a5a6a6a6a6
bbbaba2ba2babba3ba3ba4ba4ba3bba4ba3bba4ba4ba4ba4ba4
a2a2a3a2ba4a3ba4a5a6a5a6a6a6a6a6a6a6a6
ababa3a4a4a3ba5a5a6a6a6a6a6a6a6a6a6a6
bababa2babba3ba4ba3ba4ba3bba4ba3bba4ba4ba4ba4ba4ba4ba4
a3a3a4a3ba5a6a5a6a6a6a6a6a6a6a6a6a6a6
a2ba2ba4a5a5a6a6a6a6a6a6a6a6a6a6a6a6a6
ba2ba2ba3ba4ba4ba3bba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
babbabba3ba4ba4ba3bba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
a4a4a5a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6
a3ba3ba5a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6
ba3ba3ba4ba3bba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
a5a5a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6
ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
ba3bba3bba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6

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 ⇒ a6 [6]
2. ba5 ⇒ ba4 [5]
3. aba ⇒ a3 [1]
4. a4b ⇒ a6 [4]
5. b2 ⇒ ba2 [2]
6. ba2b ⇒ ba4 [3]
# ab:aba=aaa,baa=bb a/b
aaaaaaa=aaaaaa
baaaaa=baaaa
aba=aaa
aaaab=aaaaaa
bb=baa
baab=baaaa

Same cardinality

16 unique, 71 total

Σ#PresentationDescriptionRelated
104375⟨a, b | aaaa=b, abbbb=1⟩Isomorphic to ℤ1739 iso
106428⟨a, b | aab=b, babbb=a⟩Finite non-commutative monoid with 17 elements2 iso
106514⟨a, b | aba=b, baaab=a⟩Finite non-commutative monoid with 17 elements
107152⟨a, b | bb=aa, abab=aa⟩Finite non-commutative monoid with 17 elements
107153⟨a, b | bb=aa, abab=ab⟩Finite non-commutative monoid with 17 elements
107154⟨a, b | bb=aa, abab=ba⟩Finite non-commutative monoid with 17 elements
1113141⟨a, b | aab=aaa, bbb=ab⟩Finite non-commutative monoid with 17 elements
1113142⟨a, b | aab=aaa, bbb=ba⟩Finite non-commutative monoid with 17 elements
1113156⟨a, b | aba=aaa, bbb=ab⟩Finite non-commutative monoid with 17 elements
1113218⟨a, b | abb=aba, bbb=aa⟩Finite non-commutative monoid with 17 elements
1114395⟨a, b | aaaa=b, abbbb=a⟩Isomorphic to ℕ(17 = 1)2 iso
1114396⟨a, b | aaaa=b, abbbb=b⟩Isomorphic to ℕ(17 = 4)2 iso
1114698⟨a, b | aabb=a, baaaa=b⟩Finite non-commutative monoid with 17 elements1 iso, 6 anti-iso
1115488⟨a, b | aaa=ab, babbb=b⟩Finite non-commutative monoid with 17 elements3 iso
1116143⟨a, b | aab=aa, bbbb=ba⟩Finite non-commutative monoid with 17 elements
1116521⟨a, b | aab=bb, bab=aaa⟩Finite non-commutative monoid with 17 elements